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|---|---|---|
LNA simulation demonstrating two weak input signals, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi}$$\end{document}Xhi and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{hi}$$\end{document}Yhi at 6 and 4 molecules respectively, boosted by the C-element with an output \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{hi}$$\end{document}Zhi at 10 molecules
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| 99.44 |
Simulation of the final dual-rail Muller C-element design on selected input configurations. a Dual-rail AM circuit which is our final C-element design. b Deterministic simulation with inputs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{lo}, Y_{hi}$$\end{document}Xlo,Yhi at 10 molecules and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{hi}$$\end{document}Zhi at 10 molecules, which does not exhibit any change over time. c Deterministic simulation resulting in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{lo}$$\end{document}Zlo (seen in light blue) based on the initial configuration \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{lo}, Y_{lo}$$\end{document}Xlo,Ylo at 10 molecules and initial output \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{hi}$$\end{document}Zhi at 10 molecules. d LNA simulation resulting in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{hi}$$\end{document}Zhi (seen in yellow) based on the initial configuration of input \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi}, Y_{hi}$$\end{document}Xhi,Yhi at 10 molecules and initial output \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{lo}$$\end{document}Zlo at 10 molecules. Note that some plots are overlayed but are either set to 0 or 10 molecules. (Color figure online)
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other
| 97.0 |
A latch is a device used in electronics to store a logical 0 or 1, which therefore needs to have at least two stable states that are cycled between. Latches are used in asynchronous computing both for storage and for synchronisation purposes. When an input of 1 is received a latch will ideally display an output of 1, and likewise for an input of 0. We present two latch designs in Fig. 12, each intended to interface in a specific way when used within a larger system. The first simple latch, shown in Fig. 12a(i), has input \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{lo}, X_{hi}$$\end{document}Xlo,Xhi and output species \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{lo},Y_{hi}$$\end{document}Ylo,Yhi, the intuition being that either \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{lo}$$\end{document}Xlo catalyses \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{hi}$$\end{document}Yhi to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{lo}$$\end{document}Ylo or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi}$$\end{document}Xhi catalyses \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{lo}$$\end{document}Ylo to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{hi}$$\end{document}Yhi. There are also two additional reactions that catalyse \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{lo}$$\end{document}Ylo to itself and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{hi}$$\end{document}Yhi to itself, creating a feedback loop. These additional reactions ensure that, if there is a drop in the molecular concentrations of input species, the latch retains its state. For some larger systems we may need the output state of a latch to be neither \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{lo}$$\end{document}Ylo nor \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{hi}$$\end{document}Yhi, to signify that no value is stored within the latch, known as a neutral state in electronics. A reset wire, to reset a latch to neutral state, is also commonly used in circuits. To this end, the latch in Fig. 12a(ii) has an input \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{hi},R_{lo}$$\end{document}Rhi,Rlo used to reset the latch to a central state \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document}λ, as well as the standard inputs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi},X_{lo}$$\end{document}Xhi,Xlo and outputs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{hi}, Y_{lo}$$\end{document}Yhi,Ylo. The advantage of this central state, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document}λ, is that the latch can be in a state where neither \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{hi}$$\end{document}Yhi nor \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{lo}$$\end{document}Ylo are present, which is useful if these reactions are catalytic to any other component, in which case they will not be triggered directly. A comparison of the behaviour of the two latches is displayed in Fig. 12b, c. With the same initial conditions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi}$$\end{document}Xhi and output \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{lo}$$\end{document}Ylo at 10 molecules, the latch in Fig. 12a(ii) produces an output \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{hi}$$\end{document}Yhi at 10 molecules in 0.2 s. We contrast this with the latch in Fig. 12c, which outputs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{hi}$$\end{document}Yhi at 10 molecules in the slower time of 0.5 s.Fig. 12Two latch designs and their comparison. a(i) Simple latch with two feedback loops. a(ii) Latch with reset to a neutral state. b Deterministic plot of simple latch in a(i). c Deterministic plot of the latch in a(ii) which shows faster convergence
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other
| 99.6 |
An arbiter is used to decide an output signal based on which signal arrived first or if one signal is dominant over another. They are used in error correction where a signal may have degraded. Essentially, an arbiter computes the well known function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$max(|X_1|,|X_2|)$$\end{document}max(|X1|,|X2|) for two inputs. In terms of CRNs, this can be seen as either one species arriving before another or having higher molecular concentration. Since the AM circuit computes the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$max(|X_1|,|X_2|)$$\end{document}max(|X1|,|X2|) function, as one population is biased overFig. 13Arbiter circuit design and its simulation. a The arbiter CRN with inputs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi}$$\end{document}Xhi and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{lo}$$\end{document}Xlo and outputs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{hi}$$\end{document}Yhi and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{lo}$$\end{document}Ylo. The output reflects the input with the higher concentration of molecules (or which ever species appeared first). b LNA simulation of the arbiter, demonstrated with an input of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi}$$\end{document}Xhi at a concentration of 5 molecules and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{lo}$$\end{document}Xlo with a concentration of 0 molecules. After 0.4 s we see \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{hi}$$\end{document}Yhi (in red) at a concentration of 10 molecules, representing the majority input. (Color figure online) another depending upon which has the majority, it therefore serves as an appropriate candidate for an arbiter. The proposed arbiter design, seen in Fig. 13a, is the same as our AM CRN presented in Fig. 8a, except that there are two inputs, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi}$$\end{document}Xhi and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{lo}$$\end{document}Xlo, and two outputs, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{hi}$$\end{document}Yhi and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{lo}$$\end{document}Ylo, instead of four inputs. This works as desired since the output \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{hi},Y_{lo}$$\end{document}Yhi,Ylo begins to be converted from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document}λ as soon as either of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi},X_{lo}$$\end{document}Xhi,Xlo arrives, therefore automatically biasing whichever species is present first. The ability for approximate majority to reach a consensus means that this circuit can deal with stochastic fluctuations in input. Although, within electronic circuits, an arbiter outputs which signal arrived first, we assume that this information is revealed through the promotion of an output species linked to an input species.
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study
| 99.1 |
Arbiter circuit design and its simulation. a The arbiter CRN with inputs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi}$$\end{document}Xhi and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{lo}$$\end{document}Xlo and outputs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{hi}$$\end{document}Yhi and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{lo}$$\end{document}Ylo. The output reflects the input with the higher concentration of molecules (or which ever species appeared first). b LNA simulation of the arbiter, demonstrated with an input of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi}$$\end{document}Xhi at a concentration of 5 molecules and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{lo}$$\end{document}Xlo with a concentration of 0 molecules. After 0.4 s we see \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{hi}$$\end{document}Yhi (in red) at a concentration of 10 molecules, representing the majority input. (Color figure online)
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| 99.7 |
In Fig. 13b we demonstrate the operation of this arbiter by LNA simulation on inputs of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi}$$\end{document}Xhi at a concentration of 5 molecules and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{lo}$$\end{document}Xlo with a concentration of 0 molecules. After 0.4 s we see \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{hi}$$\end{document}Yhi (in red) at a concentration of 10 molecules, representing the majority.
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study
| 100.0 |
Control flow is used to mediate or propagate the flow of information throughout the computing device. In digital circuitry forks and joins, both control flow elements, are naturally implemented using wires. Unfortunately, there is no natural fork or join within CRNs and consequently we present designs for them. The fork, shown Fig. 14a, is used to split signals. It has two input species \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi}, X_{lo}$$\end{document}Xhi,Xlo and four output species, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{hi}(1), Y_{hi}(2)$$\end{document}Yhi(1),Yhi(2) to represent the splitting of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi}$$\end{document}Xhi and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{lo}(1), Y_{lo}(2)$$\end{document}Ylo(1),Ylo(2) to represent the splitting of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{lo}$$\end{document}Xlo. The join, see Fig. 14b, joins two input signals to create one output signal. There are 4 inputs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi}(1), X_{hi}(2)$$\end{document}Xhi(1),Xhi(2) with output \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{hi}$$\end{document}Yhi to represent the merging of the two input signals and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{lo}(1), X_{lo}(2)$$\end{document}Xlo(1),Xlo(2) to merge to an output \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{lo}$$\end{document}Ylo.Fig. 14CRNs for control flow components. a The fork used to split a signal into two. b The join used to merge signals
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other
| 99.8 |
Although gate designs for Boolean operators have been proposed in CRNs (Soloveichik et al. 2008), we present dual-rail implementations of logic gates in line with other designs proposed within this paper. In contrast to the gates in Soloveichik et al. (2008), our gates account for all inputs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi},X_{lo}$$\end{document}Xhi,Xlo, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{hi}, Y_{lo}$$\end{document}Yhi,Ylo and outputs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{hi}, Z_{lo}$$\end{document}Zhi,Zlo. They are also reusable and respond to changes in input.
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other
| 98.06 |
The simplest gate, NOT, in Fig. 15a(i), inverts the inputs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi}, X_{lo}$$\end{document}Xhi,Xlo to outputs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{lo}, Y_{hi}$$\end{document}Ylo,Yhi. The AND-gate, shown in Fig. 15a(iii), has initial concentrations of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _1,\lambda _2$$\end{document}λ1,λ2 as well as input concentrations. With a presence of species \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{hi}$$\end{document}Yhi we can catalyse \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _2$$\end{document}λ2 into the state \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _1$$\end{document}λ1, and with the species \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi}$$\end{document}Xhi we can catalyse \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _1$$\end{document}λ1 to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{hi}$$\end{document}Zhi; thus both species are needed for the gate to output the signal \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{hi}$$\end{document}Zhi. The state \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{hi}$$\end{document}Zhi catalyses a reaction between species \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{lo}$$\end{document}Zlo and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _3$$\end{document}λ3, therefore showing that only one output signal can be present at any time. Conversely, with either \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{lo},Y_{lo}$$\end{document}Xlo,Ylo we can convert \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _3$$\end{document}λ3 to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{lo}$$\end{document}Zlo, which in turn can convert \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{hi}$$\end{document}Zhi back to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _2$$\end{document}λ2 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _2$$\end{document}λ2 to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _1$$\end{document}λ1. Using a similar trail of thought we can see how the other gates are devised, with OR (Fig. 15a(ii)) having initial concentrations of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _2, \lambda _3$$\end{document}λ2,λ3, NOR (Fig. 15a(vi)) having initial concentrations of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _2, \lambda _3$$\end{document}λ2,λ3 and NAND (Fig. 15a(iv)) having initial concentrations of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _1, \lambda _2$$\end{document}λ1,λ2. We provide an example, showing a deterministic simulation of the AND gate, seen in Fig. 15b, in which inputs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi},Y_{hi}$$\end{document}Xhi,Yhi at 10 molecules and initial output \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{lo}$$\end{document}Zlo is converted to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{hi}$$\end{document}Zhi after 0.8 s.Fig. 15 a Dual-rail logic gate designs: we present a NOT (i), OR (ii), AND (iii), NAND (iv), XOR (v) and NOR (vi) over inputs X, Y and output Z. b Deterministic simulation for AND on inputs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi},Y_{hi}$$\end{document}Xhi,Yhi at 10 molecules and initial output \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{lo}$$\end{document}Zlo, which is converted to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{hi}$$\end{document}Zhi after 0.8 s (seen in grey)
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a Dual-rail logic gate designs: we present a NOT (i), OR (ii), AND (iii), NAND (iv), XOR (v) and NOR (vi) over inputs X, Y and output Z. b Deterministic simulation for AND on inputs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi},Y_{hi}$$\end{document}Xhi,Yhi at 10 molecules and initial output \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{lo}$$\end{document}Zlo, which is converted to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{hi}$$\end{document}Zhi after 0.8 s (seen in grey)
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XOR is slightly different. XOR, traditionally a gate that requires a composition of many other logic gates, has to be constructed with all combinations of inputs considered. The XOR gate (Fig. 15a(v)) has initial concentrations of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _1,\lambda _2,\lambda _3,\lambda _4$$\end{document}λ1,λ2,λ3,λ4. In Fig. 16 we show an example validation of the XOR gate for all four input configurations using LNA.Fig. 16 XOR gate validation demonstrated using LNA for all input combinations and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _1,\lambda _2,\lambda _3,\lambda _4$$\end{document}λ1,λ2,λ3,λ4 having initial concentrations of 10 molecules. In a we demonstrate that XOR on an input configuration \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{lo}, Y_{lo}$$\end{document}Xlo,Ylo at 10 molecules produces an output \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{lo}$$\end{document}Zlo (seen in grey) at 10 molecules after 0.4 s. d A similar plot on input \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi}, Y_{hi}$$\end{document}Xhi,Yhi, which results in output \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{lo}$$\end{document}Zlo (seen in grey). In b, c we show that both \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi}, Y_{lo}$$\end{document}Xhi,Ylo and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{lo}, Y_{hi}$$\end{document}Xlo,Yhi demonstrate the correct output of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{hi}$$\end{document}Zhi (seen in pink). (Color figure online)
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XOR gate validation demonstrated using LNA for all input combinations and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _1,\lambda _2,\lambda _3,\lambda _4$$\end{document}λ1,λ2,λ3,λ4 having initial concentrations of 10 molecules. In a we demonstrate that XOR on an input configuration \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{lo}, Y_{lo}$$\end{document}Xlo,Ylo at 10 molecules produces an output \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{lo}$$\end{document}Zlo (seen in grey) at 10 molecules after 0.4 s. d A similar plot on input \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi}, Y_{hi}$$\end{document}Xhi,Yhi, which results in output \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{lo}$$\end{document}Zlo (seen in grey). In b, c we show that both \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi}, Y_{lo}$$\end{document}Xhi,Ylo and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{lo}, Y_{hi}$$\end{document}Xlo,Yhi demonstrate the correct output of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{hi}$$\end{document}Zhi (seen in pink). (Color figure online)
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We construct a CRN to emulate the behaviour of the C-pipeline outlined in Sect. 3.2. The pipeline is a mechanism to relay handshakes between components, for example latches to store data. We construct the pipeline by placing three of our C-element CRNs, shown Fig. 11a, in series. At each intermediate stage between the C-elements we add a fork. One path of the fork is negated and fed back into the previous C-element, and the other path is fed into the new C-element. The key interaction between the components of the C-pipeline is that no C-element can output a positive species or negative species without the previous displaying a positive or negative one.
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The inputs to the C-pipeline CRN are \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Req_{hi}, Req_{lo}, Acc_{hi}$$\end{document}Reqhi,Reqlo,Acchi and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Acc_{lo}$$\end{document}Acclo. The only output is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{hi}, C_{lo}$$\end{document}Chi,Clo, which is the output species of the third C-element. However, for the sake of clarity, we also include four other species \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{hi}, A_{lo}$$\end{document}Ahi,Alo corresponding to the output of the first C-element and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{hi}, B_{lo}$$\end{document}Bhi,Blo corresponding to the second. On an input of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Req_{hi}$$\end{document}Reqhi at 10 molecules we would expect to see that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{hi}$$\end{document}Ahi, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{hi}$$\end{document}Bhi and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{hi}$$\end{document}Chi are all at 10 molecules after a staggered amount of time. If we then changed the input to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Req_{lo}$$\end{document}Reqlo we would expect to see \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{hi}$$\end{document}Ahi, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{hi}$$\end{document}Bhi, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{hi}$$\end{document}Chi diminish with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{lo}$$\end{document}Alo, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{lo}$$\end{document}Blo, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{lo}$$\end{document}Clo, reaching 10 molecules to reflect the change in input. This is seen as a ‘wave’ through the pipeline propagating a high signal and then a low signal.
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We design an experiment, see Fig. 17, where we initialise the pipeline with the input species \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Req_{hi}$$\end{document}Reqhi at 10 molecules, and all C-elements are initialized with the intermediary species \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document}λ at 10 molecules such that no C-element yet outputs a species. From both the deterministic and LNA simulations of this we can observe how the species \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{hi}, B_{hi}, C_{hi}$$\end{document}Ahi,Bhi,Chi approach 10 molecules one after another, showing that indeed the value of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Req_{hi}$$\end{document}Reqhi is being propagated along the pipeline. In order to show that our pipeline design is continuously reactive, we convert all of the species \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Req_{hi}$$\end{document}Reqhi to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Req_{lo}$$\end{document}Reqlo via the introduction of a reaction \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Req_{hi} \rightarrow Req_{lo}$$\end{document}Reqhi→Reqlo. This effect occurs at around 1 s and we can observe that the pipeline responds by reducing the species \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{hi}, B_{hi}, C_{hi}$$\end{document}Ahi,Bhi,Chi to a molecular count of 0. We also observe (not shown on the simplified plot) that the species \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{lo}, B_{lo}, C_{lo}$$\end{document}Alo,Blo,Clo reach 10 molecules at the same time as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{hi}, B_{hi}, C_{hi}$$\end{document}Ahi,Bhi,Chi reach 0 molecules (2 s). The LNA plot reveals that all output species are separated by a significant time difference such that no two species can be conflated.Fig. 17Validation of the Muller C-pipeline. The input request signal, encoded by the species \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Req_{hi}$$\end{document}Reqhi, is propagated to the end of the pipeline (represented by the species \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{hi}, B_{hi}, C_{hi}$$\end{document}Ahi,Bhi,Chi); we then set the request signal to low. The pipeline then responds by the presence of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{hi}, B_{hi}$$\end{document}Ahi,Bhi and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{hi}$$\end{document}Chi diminishing to zero. In b we show that the variance is low, even for low molecular counts
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Validation of the Muller C-pipeline. The input request signal, encoded by the species \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Req_{hi}$$\end{document}Reqhi, is propagated to the end of the pipeline (represented by the species \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{hi}, B_{hi}, C_{hi}$$\end{document}Ahi,Bhi,Chi); we then set the request signal to low. The pipeline then responds by the presence of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{hi}, B_{hi}$$\end{document}Ahi,Bhi and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{hi}$$\end{document}Chi diminishing to zero. In b we show that the variance is low, even for low molecular counts
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In addition to simulation experiments which plot expected concentrations of species over time, we also check temporal properties concerning the interactions between species and components. Using the PRISM model checker we interrogate the CTMC models of the pipeline with specific queries. We give some important examples of such queries in Table 1, which are verified by PRISM as being true with very high probability by checking 20 paths against the property. “Th” refers to the required population threshold which can be set by the user. PRISM also has the ability to track and plot concentrations of species over specific time intervals and number of samples. For example, for the C-pipeline we may wish to focus specifically on species concentrations of the second C-element whilst ignoring the others. We can isolate the species in question starting at a time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$> 0$$\end{document}>0 and simulating only the species of the second C-element. We demonstrate this property on the pipeline with initial input species \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Req_{hi}$$\end{document}Reqhi at 10 molecules in Fig. 18.Table 1Temporal properties for the C-pipeline verified by the PRISM model checker. Each property was checked on 20 paths for the pipeline with inputs at 10 moleculesProperty in EnglishInitial conditionPRISM queryProb. of Success“Probability that the first C-element always outputs a high signal before the second within 3 s” \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Req_{hi} > Th$$\end{document}Reqhi>Th \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P=?[(B_{hi} < Th) {\mathcal {U}}^{[0 , 3]}$$\end{document}P=?[(Bhi<Th)U \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(A_{hi} > Th)]$$\end{document}(Ahi>Th)] 0.97“Probability that the C-element only changes when both inputs change within 10 s”(1) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Req_{hi} Acc_{lo}$$\end{document}ReqhiAcclo (2) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Req_{lo} Acc_{hi}$$\end{document}ReqloAcchi (3) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Req_{hi} Acc_{hi}$$\end{document}ReqhiAcchi \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P=?[\text {true }{\mathcal {U}}^{} Z_{hi} > Th ]$$\end{document}P=?[trueUZhi>Th] 0.96“Probability that the species \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{hi}$$\end{document}Ahi reach their maximum population within 10 s” \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{hi} < Th$$\end{document}Ahi<Th \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P=?[\text {true } {\mathcal {U}}^{} A_{hi} >= Th]$$\end{document}P=?[trueUAhi>=Th] 1“Probability that request signal is propagated to the end of the pipeline within 10 s” \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Req_{hi} > Th$$\end{document}Reqhi>Th, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{hi}> Th$$\end{document}Ahi>Th \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P=?[F^{} C_{hi} > Th]$$\end{document}P=?[FChi>Th] 1
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Fig. 18The expected concentration of species \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{hi}$$\end{document}Bhi, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{lo}$$\end{document}Blo and intermediary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document}λ plotted over time for the C-pipeline under the stochastic semantics using reward structures within PRISM. Each data point is the expectation over 20 samples. We start at t = 0.2. With an initial condition of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Req_{hi}$$\end{document}Reqhi, we can see (in blue) that species \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{hi}$$\end{document}Bhi is output from the second C-element at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t > 1.25$$\end{document}t>1.25. (Color figure online)
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| 99.94 |
The expected concentration of species \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{hi}$$\end{document}Bhi, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{lo}$$\end{document}Blo and intermediary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document}λ plotted over time for the C-pipeline under the stochastic semantics using reward structures within PRISM. Each data point is the expectation over 20 samples. We start at t = 0.2. With an initial condition of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Req_{hi}$$\end{document}Reqhi, we can see (in blue) that species \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{hi}$$\end{document}Bhi is output from the second C-element at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t > 1.25$$\end{document}t>1.25. (Color figure online)
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| 99.9 |
We have also designed and validated a queue, the schematic for which is shown in Fig. 19, built by the addition of latches at each C-element block to the Muller pipeline. The queue is used in electronics to regulate and store the flow of information. The asynchronous queue uses the pipeline as a control mechanism to propagate signals between the latches. We use the latch with reset seen in Fig. 12 for this purpose. As the species \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Req_{hi}$$\end{document}Reqhi is propagated along the pipeline, it sends a signal to the queue to read and store the value in the next latch along. Each latch represents some computation that could be completed within each time interval.
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| 99.6 |
For the latches of our queue we have input species \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Am_{hi}$$\end{document}Amhi and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Am_{lo}$$\end{document}Amlo and output species \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Ams_{hi}, Bms_{hi},Cms_{hi}$$\end{document}Amshi,Bmshi,Cmshi representing the output of each latch. Deterministic simulation the queue pipeline is shown in Fig. 19. In this experiment we propagate a 1 (represented as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Am_{hi}$$\end{document}Amhi at 10 molecules) followed by 0 (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Am_{lo}$$\end{document}Amlo at 10 molecules). We can observe the species \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Ams_{hi}, Bms_{hi}$$\end{document}Amshi,Bmshi, which represent the outputs of the first and second latches, noting an oscillatory pattern of cycling between 1 and 0.Fig. 19Deterministic simulation of the queue pipeline. We propagate a value of 1 through the queue. The species \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Ams_{hi}, Bms_{hi}$$\end{document}Amshi,Bmshi represent the outputs of the first and second latches. Note that through oscillatory patterns generated by the pipeline we can mimick properties of a synchronous system
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| 76.0 |
Deterministic simulation of the queue pipeline. We propagate a value of 1 through the queue. The species \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Ams_{hi}, Bms_{hi}$$\end{document}Amshi,Bmshi represent the outputs of the first and second latches. Note that through oscillatory patterns generated by the pipeline we can mimick properties of a synchronous system
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| 99.9 |
We have also designed a three bit ripple-carry adder, which operates in a similar fashion to the queue but instead of latches we compose adders in series, seen in Fig. 20b. An adder, the circuit design for which is seen in Fig. 20a, is a composition of two XOR gates, two OR gates and an AND gate. The adder produces two outputs, the value of the summation and the carry. Within the ripple-carry adder, the carry output of each adder is fed into the next adder, which outputs the sum and a carry. In this way, with three adders, we can add three two-bit numbers together.Fig. 20 a Circuit diagram for a ripple-carry adder. b Three adders in series controlled by the C-pipeline. A carry-bit output from one adder is fed into the next as part of the input
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| 99.9 |
Our ripple-carry CRN has four input species per adder representing the two inputs, and two output species representing the output. In Fig. 21 we show that the adder exhibits correct behaviour by producing the desired output species for a specific input, where each sum is calculated only in the next stage in the pipeline. If we view each C-element and adder as one stage in the pipeline, labelled A, B and C, then we can view the output species of each adder as a bridge to the next adder. The six output species, representing the carry-bit output, are denoted by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{abridgeOneOut}, B_{bbridgeOneOut}, C_{cbridgeOneOut}$$\end{document}AabridgeOneOut,BbbridgeOneOut,CcbridgeOneOut and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{abridgeZeroOut}, B_{bbridgeZeroOut}, C_{cbridgeZeroOut}$$\end{document}AabridgeZeroOut,BbbridgeZeroOut,CcbridgeZeroOut. In order to show correct operation the output of the adders (represented in this case by 10 molecules) should be interleaved with the control species of the C-pipeline (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{hi},B_{hi},C_{hi}$$\end{document}Ahi,Bhi,Chi), allowing time for the carry species to catalyse with the input of the following adder.Fig. 21Deterministic simulation of the ripple-carry adder circuit responding to various inputs. We plot the output species from each section of the pipeline used to coordinate the output from each adder. The carry-bit output from each adder is represented by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{abridgeOneOut}, B_{bbridgeOneOut}$$\end{document}AabridgeOneOut,BbbridgeOneOut and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{cbridgeOneOut}$$\end{document}CcbridgeOneOut. The output of the C-element (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{hi},B_{hi},C_{hi}$$\end{document}Ahi,Bhi,Chi) arrives strictly before the output from the adder. The logical input for a is 1 and 0 for the first adder, 1 and 0 for the second adder, and 1 and 0 for the third adder. In b we show the computation on different inputs, namely 1 and 0 for the first adder, 0 and 0 for the second adder, and 1 and 0 for the third adder. The crossover in the concentrations of output species of the C-element and the logical output of 1 (resulting from inputs 0, 1 and carry of 1) in the third adder (plots \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{hi}$$\end{document}Chi and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{csOneOut}$$\end{document}CcsOneOut) indicates faster convergence but does not affect the results in further stages of the pipeline
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| 97.7 |
Deterministic simulation of the ripple-carry adder circuit responding to various inputs. We plot the output species from each section of the pipeline used to coordinate the output from each adder. The carry-bit output from each adder is represented by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{abridgeOneOut}, B_{bbridgeOneOut}$$\end{document}AabridgeOneOut,BbbridgeOneOut and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{cbridgeOneOut}$$\end{document}CcbridgeOneOut. The output of the C-element (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{hi},B_{hi},C_{hi}$$\end{document}Ahi,Bhi,Chi) arrives strictly before the output from the adder. The logical input for a is 1 and 0 for the first adder, 1 and 0 for the second adder, and 1 and 0 for the third adder. In b we show the computation on different inputs, namely 1 and 0 for the first adder, 0 and 0 for the second adder, and 1 and 0 for the third adder. The crossover in the concentrations of output species of the C-element and the logical output of 1 (resulting from inputs 0, 1 and carry of 1) in the third adder (plots \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{hi}$$\end{document}Chi and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{csOneOut}$$\end{document}CcsOneOut) indicates faster convergence but does not affect the results in further stages of the pipeline
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| 99.3 |
Whilst a simulation provides the intuition behind the ripple-carry adder, using the PRISM model checker we can query the output of all three adders after 10 s to confirm if the correct output is present. We have four input species per adder, excluding the carry, which represent two numbers. We expect one species from each adder as output, representing the addition of two inputs, plus the carry. The third adder relies on the previous adder’s carry being correct. We therefore only need to look at the desired output of each adder plus the carry of the final adder. We summarise this with the following example PRISM property:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&P =? [ \text { true } U <= 10 ( (A_{abridgeOneOut}> Th) \text { and }\\&(B_{bbridgeZeroOut}> Th) \text { and } ( C_{cbridgeOneOut}> Th) \text { and }\\&(C_{cbridgeCarryOneOut} > Th) ) ] \end{aligned}$$\end{document}P=?[trueU<=10((AabridgeOneOut>Th)and(BbbridgeZeroOut>Th)and(CcbridgeOneOut>Th)and(CcbridgeCarryOneOut>Th))]With this example we can only satisfy these three outputs and the carry, by seeing each of their molecular concentrations rise above the threshold Th, based on a specific input configuration. This particular input are the species representing 0 and 1 for the first adder (for example \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{aOneZeroIn}$$\end{document}AaOneZeroIn and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{aTwoOneIn}$$\end{document}AaTwoOneIn), the species representing inputs 1,1 for the second adder and 1,1 for the third adder. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 + 1$$\end{document}0+1 in the first adder should give us an outcome of 1 carry 0 and so satisfies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{abridgeOneOut} > Th$$\end{document}AabridgeOneOut>Th. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 + 1$$\end{document}1+1 plus the 0 carry from the first adder gives an output of 0 carry 1, and so satisfies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{bbridgeZeroOut} > Th$$\end{document}BbbridgeZeroOut>Th. An input of 1 + 1 plus 1 carry from the second adder means that our output should be 1 as well as the final carry should be 1, represented by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{cbridgeOneOut} > Th$$\end{document}CcbridgeOneOut>Th and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{cbridgeCarryOneOut} > Th$$\end{document}CcbridgeCarryOneOut>Th. Our adder satisfies this property based upon the inputs given and therefore shows correct operation for an adder.
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| 91.2 |
All designs3 presented in this paper have been validated using both Microsoft’s Visual GEC tool (Pedersen and Phillips 2009) and the PRISM model checker (Kwiatkowska et al. 2011), both for the deterministic and stochastic semantics of the CRNs. Visual GEC provides a programming language, LBS, for designing and simulating any given CRN. We systematically tested each component in isolation by simulating its behaviour against all input and output configurations. Next, we examined how a component might behave in a larger system, where it will be exposed to a change in input. To this end, we introduced new reactions to emulate a signal change. For instance, if we wished to change a carrier signal from high to low, we would introduce an additional reaction \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi} \overset{k}{\rightarrow } X_{lo}$$\end{document}Xhi→kXlo, which converts all of the signal \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi}$$\end{document}Xhi into a signal \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{lo}$$\end{document}Xlo while the component is operating.
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| 99.94 |
Since deterministic semantics is not accurate for low molecular populations, we additionally explored stochastic semantics. Visual GEC exports models to the probabilistic model checker PRISM, which then enables verification of the induced continuous-time Markov chain against temporal logic properties. This allows one to check that the circuits ensure the correct temporal ordering of the events, for example, for the Muller pipeline seen in Fig. 6, that the species in the first stage of the pipeline is present before the species in the second, i.e. with probability 1, and that the signal is eventually propagated to the end of the pipeline. PRISM implements numerical solution of the CME, which is exponential in the initial number of molecules and hence not scalable, and analysis based on stochastic simulation, which is time consuming. We thus additionally used an experimental implementation of the LNA within Visual GEC, based on Cardelli et al. (2015). As well as being capable of checking temporal logic properties (Cardelli et al. 2015; Bortolussi et al. 2016), the LNA can plot the species concentration over time together with standard deviation, and is fast and reasonably accurate even for low molecule counts. Moreover, compared to the deterministic semantics, LNA provides important information about stochasticity that may affect the robustness of the circuits, and which can be explored further with CME, stochastic simulation, or verifying that the circuit converges with probability 1 to a single value.
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| 100.0 |
When modelling asynchronous circuits as a chemical system, the wires are chemical species from the output set of one gate component to the input set of another. We cannot bound the time for a gate to transform an input species to an output species. This excludes the class of self-timed circuits. Under deterministic semantics, we could guarantee an isochronous fork since two chemical species, either high or low, could theoretically reach the threshold M at precisely the same time given equal rates and initial concentrations, and therefore under deterministic semantics we have a Turing-complete method of computation. We cannot guarantee this under stochastic semantics (Cook et al. 2009). This is because there is a non-zero probability that one species could reach M before the other. We thus conclude that our circuits, at worst, can be classified as speed independent. We can calculate approximately the delay on wires based upon rates and concentrations for each semantics.
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study
| 98.44 |
Direct chemical implementations of CRNs have been theorised and realised, but involve complicated reaction mechanisms (Shin 2011). The most common substrate for chemical kinetics is DNA strand displacement (DSD), which involves the displacement of DNA strands in solution. These strands are labelled with the chemical species and, once the reaction has taken place, an outputting strand represents an output from the CRN that the strand displacement system is trying to emulate. DNA strand displacement has been shown to be a universal substrate for chemical kinetics, specifically for bi-molecular reactions used here (Soloveichik et al. 2010). Most importantly, the AM circuit seen in Fig. 8b has been implemented as a strand displacement device (Lakin et al. 2012). However, a potential difficulty with this approach is scalability: as the number of components increases, the number of chemical species representing them also increases. Large numbers of chemical species result in large numbers of DSD complexes in solution, and consequently crosstalk needs to be accounted for. Fortunately, a recent experiment with the implementation of a square-root circuit in solution provided a new experimental ceiling on the number of species that can be used (Qian and Winfree 2011).
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study
| 100.0 |
Another challenge is to provide formal verification of the correctness of the designs. We remark that, although we have validated the behaviour of the components for all possible input configurations and verified that correct outputs are produced and in the correct order using simulation and simulation-based probabilistic model checking with PRISM, this does not amount to full verification. Asynchronous diagrams are represented using a variety of notations, see Fig. 5, and correspond to certain classes of (safe) Petri nets (Myers 2004) known as Signal Transition Graphs (STGs), representing the rise and fall of signals. Our designs are systems containing many molecules which exhibit stochastic behaviour. Firstly, one would need to show that our CRN designs meet the specification given as a Petri net, which would involve relating the two formally via a refinement relation, where one needs to relate structures with many molecules of a given a species to structures with at most one. This presents us with two major challenges: scalability and stochasticity. Scalability can be addressed using compositional verification, which has been developed for (non-probabilistic) process algebraic specifications of asynchronous circuits (Wang and Kwiatkowska 2007) (equivalent to STGs) and it would be interesting to see if they can be applied in this setting. However, no probabilistic extension of this approach is known. Another possibility is to capture stochasticity by employing stochastic Petri nets to model the designs as done for molecular walkers in Barbot and Kwiatkowska (2015), and then perform temporal logic verification using the tool Cosmos. Cosmos relies on an implementation of statistical model checking that exploits parallelism of the Petri net specifications and achieves greater scalability than PRISM.
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study
| 99.94 |
We have proposed a novel design for an asynchronous computing device based on Chemical Reaction Networks. CRNs are inherently asynchronous, and thus particularly well suited to this computational paradigm. Our designs are based on a simple, bi-molecular reaction motif inspired by Approximate Majority (Angluin et al. 2008; Cardelli and Csikász-Nagy 2012), employ catalytic reactions and assume well-mixed solution and constant, uniform rates. Moreover, they do not rely on the universal clock which is difficult to realise. Since an arbitrary CRN can be physically realised using DNA strand displacement (Soloveichik et al. 2010), as recently demonstrated experimentally in Chen et al. (2013), the proposed designs are in principle implementable, and we have confirmed this in theory by modelling them in the two-domain setting (Cardelli 2010) using Visual DSD (Phillips and Cardelli 2009; Lakin et al. 2011). Our designs are the first feasible implementation of an asynchronous computing device in chemical kinetics and are relevant for a multitude of applications in nanotechnology and synthetic biology. As future work we would like to investigate alternative experimental settings.
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study
| 99.94 |
It has long been recognized that individuals differ in perceptual sensitivity (Nebylitsyn et al., 1960; Eysenck, 1967). The term sensitivity, however, has multiple meanings. On the one hand sensitivity can be regarded as lower threshold for weak stimuli; on the other hand sensitivity can be conceived as low tolerance or overreactivity to strong stimulation. According to several theorists (Nebylitsyn et al., 1960; Eysenck, 1967; Aron and Aron, 1997) both types of sensitivity arise from the same trait. Thus, individuals with a low perceptual threshold will also have a low level of tolerance for strong stimulation. However, it has also been argued that these two types of sensitivity are independent (Ellermeier et al., 2001; Evans and Rothbart, 2008). To shed new light on this debate we examined the relation between these tendencies within the Predictive and Reactive Control Systems (PARCS) theory (Tops et al., 2010, 2014).
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study
| 99.9 |
Predictive and Reactive Control Systems theory differentiates two types of reactivity: punishment reactivity and reward reactivity. High punishment reactivity to strong stimuli corresponds to low tolerance for strong stimulation such as noises, light flashes, and odors, and a tendency to experience negative affect from it. High reward reactivity to strong stimuli corresponds to a tendency to derive pleasure from strong stimulation. In PARCS theory these two temperamental tendencies overlap in terms of high reactivity toward stimuli in the environment, but oppose each other in terms of the response orientation (approach or avoid) toward these stimuli. Due to this opposing relationship each type of reactivity can suppress [statistically; see MacKinnon et al. (2000)] the relation between the other type of reactivity and sensitivity to weak stimuli. In the present study we included measures of sensitivity to weak stimuli and of both types of reactivity to be able to test the predicted suppression effects that follow from PARCS theory. This allowed us to investigate whether PARCS theory provides a suitable framework to better understand the dependencies between perceptual sensitivity to weak stimuli and reactivity to strong stimuli.
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study
| 100.0 |
Before proceeding to the methodological details of our study we will briefly review theories on sensitivity to weak stimuli and reactivity to strong stimuli and evidence supporting these theories. First we will discuss theories that consider sensitivity to weak stimuli and reactivity to strong stimuli as one trait. Then we will discuss theories that regard both traits as independent. Finally we will discuss PARCS theory and our predictions regarding sensitivity and reactivity based on PARCS theory.
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review
| 99.9 |
One influential theory that regards sensitivity to weak stimuli and overreactivity to strong stimuli to result from one underlying trait is H. J. Eysenck’s personality theory about extraversion and introversion (Eysenck, 1967). According to Eysenck, introverts have higher arousal levels than extraverts, which causes higher cortical excitability in introverts. Due to their higher cortical excitability, introverts respond more strongly to stimulation than extraverts and have lower thresholds for weak stimulation, but they are also more easily over-aroused by strong stimulation. Because each individual tries to maintain an optimal arousal level, introverts are predicted to seek non-arousing (social) situations, while extraverts seek situations that are highly arousing (Eysenck, 1967). Questionnaires, such as the Eysenck Personality Questionnaire (EPQ; Eysenck and Eysenck, 1975), are based on this prediction and assess the level of Extraversion-Introversion through questions about (social) strategies to maintain optimal arousal level.
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review
| 98.6 |
A few early empirical studies related Eysenck’s introversion-extraversion personality dimension to sensory sensitivity as measured by threshold performance. Smith (1968) found that higher introversion was indeed associated with lower auditory thresholds for low frequency tones. In addition, Siddle et al. (1969) obtained a significant negative association between extraversion and visual sensitivity as measured by the inverse of the lower absolute threshold. However, neuroticism, another of Eysenck’s personality dimensions, was suggested to be a confounding variable in this study, making it difficult to conclude whether sensitivity arises from extraversion, neuroticism, or from a combination of these two traits. A further limitation of both these studies was that the performance on the psychophysical measures used may have not only depended on actual perceptual sensitivity but also on the criterion for responding. In terms of signal detection theory (Green and Swets, 1966; Macmillan and Creelman, 2005), the criterion for responding reflects how strong the internal signal (e.g., the sensory effect produced by a stimulus) needs to be for an individual to decide that a signal is present. The criterion an individual adopts can differ strongly between situations and tasks and depends, for example, on signal probability or the relative value of correctly detecting or correctly rejecting a signal and the relative cost of missing a signal and falsely reporting a signal. Therefore, Edman et al. (1979) carried out a study using a threshold procedure developed to measured sensitivity independent of the response criterion. They found that introverts had lower detection thresholds for electrocutaneous stimulation, but only when they also scored high on neuroticism. Taken together, these studies provide some support for the idea that sensitivity arises from an underlying trait that may also give rise to over-arousal by strong stimulation. However, the personality questionnaires used in these studies only included questions about (social) strategies to maintain optimal arousal and did not ask about over-arousal by strong stimulation directly. This makes it difficult to draw strong conclusions from these studies concerning the relation between sensitivity to weak stimuli and reactivity to strong stimuli.
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review
| 99.7 |
Another, more recent, theory that regards sensitivity and overreactivity as belonging to one trait is the highly sensitive person (HSP) theory developed by Aron and Aron (1997). Central to this theory is the trait sensory processing sensitivity (SPS). SPS is regarded as an evolutionary beneficial survival strategy. It is characterized by heightened awareness of subtle external and internal stimuli. While beneficial in certain situations, this trait comes with the cost of getting more easily overwhelmed by stimulating or quickly changing environments (Aron and Aron, 1997; Aron et al., 2012). Aron and Aron (1997) developed the HSP scale as a unidimensional scale to assess SPS. In line with the presumption that sensitivity and reactivity arise from the same trait, the HSP scale includes items that measure sensitivity to subtle stimuli as well as the tendency to get overwhelmed by strong stimulation. However, in a critical analysis of the HSP scale Evans and Rothbart (2008) questioned the unidimensionality of the scale and argued that sensitivity to weak stimulation and overreactivity are independent traits.
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review
| 99.0 |
To test the unidimensionality of the HSP scale Evans and Rothbart (2008) carried out factor analysis on the HSP scores taken from a sample of 297 undergraduates and compared the factor scores with scores on several scales of the Adult Temperament Questionnaire (ATQ; Evans and Rothbart, 2007). The ATQ is based on a multidimensional approach to temperament that subdivides each central temperamental trait into several sub-traits. It enables fine-grained exploration of relationships between these traits (Derryberry and Rothbart, 1988). Importantly, the questionnaire includes separate scales for assessing sensitivity to low-intensity stimulation and perceptual discomfort (overreactivity) due to high-intensity stimulation. Definitions of these ATQ (sub)scales can be found in Table 1.
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| 100.0 |
Factor analysis on the HSP items indicated that the HSP scale consisted of two separate factors. The first factor was strongly associated with ATQ negative affectivity and its discomfort subscale in particular. The other factor correlated highly with ATQ orienting sensitivity and its neutral perceptual sensitivity subscale (Evans and Rothbart, 2008). These results do not support the unidimensionality of the HSP scale. Furthermore Evans and Rothbart (2008) did not find a relationship between neutral perceptual sensitivity and discomfort, between orienting sensitivity and discomfort, or between the two factors of the HSP scale. The absence of these relationships questions the unidimensional view that individuals with high sensitivity to weak stimuli also have high reactivity to strong stimulation.
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| 100.0 |
Other findings that conflict with the unidimensional view were reported by Ellermeier et al. (2001). In a group of 61 volunteers they investigated the idea that increased reactivity to noise in the environment is (partly) due to increased auditory acuity. To measure reactivity to noise they used a psychometrically evaluated noise sensitivity questionnaire. Note that the term noise sensitivity in this study referred to a stable personality trait concerning perceptual, cognitive, affective and behavioral reactivity toward environmental noises. Noise sensitivity as measured by this questionnaire does thus not refer to sensitivity to weak stimuli, but can be regarded as measure of reactivity (or discomfort in terms of the ATQ) in the auditory domain. Auditory acuity was measured using several measures, including an adaptive forced-choice measure of the absolute threshold of hearing, which can be regarded an objective psychophysical measure of sensitivity to low intensity stimulation. This measure is of specific interest here because it is similar, although methodologically improved, compared to the measures used in the empirical studies discussed above (Smith, 1968; Siddle et al., 1969; Edman et al., 1979) that found relations between perceptual sensitivity and extraversion. Ellermeier et al. (2001), however, found no significant relationship between their measure of reactivity to noise and auditory acuity, including the threshold of hearing. In line with the conclusions of Evans and Rothbart (2008), this finding suggests that reactivity and sensitivity, at least in the auditory domain, are independent from each other and are not originating from a single trait.
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| 99.0 |
Taken together, there is some evidence supporting the claim that sensitivity to weak stimuli and overreactivity to strong stimuli arise from the same trait (Smith, 1968; Siddle et al., 1969; Edman et al., 1979). However, other research findings suggest that sensitivity to weak stimuli and overreactivity are independent traits (Ellermeier et al., 2001; Evans and Rothbart, 2008). There is thus disagreement about the relationship between sensitivity to weak stimuli and overreactivity to strong stimuli. A solution may be found in PARCS theory (Tops et al., 2010, 2014). In the next paragraphs we will briefly set out PARCS theory and its predictions regarding the relationship between sensitivity to weak stimuli and overreactivity to strong stimuli. Because in the present study we operationalized constructs of PARCS theory using ATQ scales (Evans and Rothbart, 2007), we will relate these constructs to the labels used in the adult temperament model by Derryberry and Rothbart (1988), Evans and Rothbart (2007) when we introduce them.
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| 94.25 |
Predictive and Reactive Control Systems theory provides an integrative framework for understanding psychological states and traits based on functioning of two major control systems in the brain: the predictive and reactive control system. These systems regulate cognition, autonomic responses, behavior, homeostasis, and emotion. PARCS theory describes temperamental or personality traits as dispositional bias toward the reactive or toward the predictive control systems, which are each adaptive in specific environments and contexts. Predictive temperament, which we will also refer to as high predictivity, is characterized by dispositional bias toward the predictive system. The predictive system controls behavior based on internal models that predict which actions will be effective for reaching goals in a given context and allows planning for future events. Predictive temperaments likely evolved to be adaptive in predictable environments, and predictive control is still deemed adaptive in such environments in modern day life (Tops et al., 2010, 2014). For example, when driving in a familiar city with organized traffic where traffic rules are obeyed one can adopt a largely feedforward approach, following previously learned rules and habits applicable to the current context, and plan behavior based on predictions about the future (such as planning ahead what is the best route to arrive home on time while passing by the cheapest gas station and do the weekly groceries on the way). In such a situation a predictive temperament is thus advantageous. In contrast to predictive temperaments, reactive temperaments are characterized by dispositional bias toward the reactive systems, which control behavior in a momentary fashion through feedback from the continuous stream of external stimuli. Reactive control is adaptive in novel, unpredictable and unstable environments (Tops et al., 2010, 2014). For example when driving for the first time in a foreign city with disorganized, busy traffic where other drivers do not (seem to) comply with traffic rules, one needs to adopt a feedback guided strategy, be constantly vigilant to environmental stimuli, and ready to immediately respond to the rapidly and unexpectedly changing situational demands (such as a car ending up in front of you after suddenly changing multiple lanes). In this situation a reactive temperament, which we will also refer to as high reactivity, is thus more suitable than a predictive temperament.
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| 99.8 |
Crucial to the current study, PARCS theory distinguishes two types of reactive systems: the reactive avoidance system and the reactive approach system. In line with other biopsychological theories of personality and temperament based on research in humans (e.g., Cloninger et al., 1993; Corr, 2004; Evans and Rothbart, 2007; cf. Gray and MacNaughton, 2000), the discrimination of the reactive approach and avoidance systems is inspired on the model of anticorrelated reward- and punishment systems developed by Gray (1970, 1989) on the basis of animal research. Individuals with bias toward the reactive avoidance system have a strong drive to process (potentially) aversive stimuli and experience aversion in order to avoid these stimuli. This drive is expressed as elevated anxiety and harm avoidance (Pickering and Gray, 1999). We will refer to dispositional bias toward the reactive avoidance system as high punishment reactivity. In the adult temperament model developed by Derryberry and Rothbart (1988), Evans and Rothbart (2007) this construct is labeled as negative affect. In addition to fear, sadness and frustration it encompasses discomfort, which, as discussed above reflects aversive responding to strong stimuli (Derryberry and Rothbart, 1988; Evans and Rothbart, 2007). Discomfort can thus be understood as punishment reactivity specifically toward strong sensory stimulation. On the other hand, individuals with bias toward the reactive approach system have a strong drive to process (potentially) appetitive stimuli in order to approach these stimuli. This drive is expressed as elevated reward responsiveness and sensation seeking (Pickering and Gray, 1999). We will refer to dispositional bias toward the reactive approach system as high reward reactivity. This construct is labeled extraversion/surgency in Derryberry and Rothbart (1988), Evans and Rothbart (2007) temperament model. Besides sociability and positive affect it includes high intensity pleasure, which reflects the tendency to derive pleasure from strong stimuli (Derryberry and Rothbart, 1988; Evans and Rothbart, 2007). High intensity pleasure can thus be regarded as reward reactivity specifically toward strong sensory stimulation. According to PARCS theory, depending on the dispositional bias toward the reactive approach or avoidance system one has, the same stimulus may be experienced differently. When confronted with strong stimuli such as loud music, individuals with high punishment reactivity will have a tendency to experience these stimuli as aversive and to be avoided, i.e., as punishment. By contrast, individuals with high reward reactivity will have a tendency to experience these stimuli as pleasurable and to be approached, i.e., as reward.
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| 99.8 |
Predictive and Reactive Control Systems theory builds on, and somewhat reorganizes the above theories of reactive approach and avoidance systems, additionally based on evidence from brain lesion and neuroimaging studies in humans (Tops et al., 2010, 2014, 2016). Furthermore, as mentioned earlier, PARCS theory adds a predictive system to the model. In contrast to the reactive system that enables immediate, mutually incompatible avoidance and approach responding to novel, urgent punishment and reward stimuli, the predictive system utilizes internal models of effective ways to respond to familiar stimuli and contexts. Those internal models represent relationships between entities, motivations, actions, and outcomes and are formed by prior learning during exposure to similar stimuli (Quirin et al., 2015). When the individual encounters similar situations in the future, integrated experiences stored in the internal model can be recalled and will provide context and perspectives for perception and appraisal of the situation and potential actions. Note that this can also apply to punishments or rewards that have been previously integrated into internal models. When presented with a previously integrated compared to a novel punishment or reward stimulus an individual can more readily and flexibly switch from reactive control, with its narrow focus on the salient stimulus, to predictive control, which is less emotionally reactive and more mindful and provident in nature (Tops et al., 2014; Quirin et al., 2015).
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| 99.9 |
Although the reactive system is specifically equipped for immediate responding to stimuli in rapidly changing environments, this does not mean that it operates without higher levels of cognitive processing. According to PARCS theory both the predictive and the reactive system have attentional and cognitive control functionality, but each has a different mode of processing. Reactive control is feedback guided and includes processes such as orienting, appraisal (i.e., assessment of stimuli in the environment on significance for well-being; Moors et al., 2013), working memory maintenance, and actively sustained attention such as needed for detection of infrequent stimuli. Predictive control, on the other hand, works in a feed-forward fashion, including processes such as planning for future events and inductive reasoning (Tops et al., 2014). This distinction in reactive and predictive cognitive processes is also reflected in neuroimaging and anatomical data. Cortical areas of the reactive system that regulate reactive reward and punishment systems (Tops et al., 2014), such as the anterior insula (AI) and dorsal anterior cingulate cortex (dACC), receive many projections from limbic and subcortical areas of the reactive punishment and reward systems such as the amygdala and ventral striatum. By contrast, cortical areas of the predictive control system, such as the posterior cingulate cortex and precuneus, receive less such projections but seem to downregulate those areas (Devinsky et al., 1995; Tops and Boksem, 2012).
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| 98.3 |
Integrating the above and other evidence, PARCS theory (Tops et al., 2010, 2014, 2016) suggests that the reactive system and the predictive system tend to inhibit each other, producing anticorrelated activation. At the same time, within the reactive system, the approach and avoidance systems also tend to inhibit each other (Gray, 1970, 1989). Accordingly, reward and punishment reactivity have in common that both reflect reactive, rather than predictive, temperaments, as both are mediated by reactive systems. At the same time these temperaments oppose each other because, in immediate, reactive action control, each reflects a different action orientation (approach or avoid). Thus, PARCS theory predicts that both reward and punishment reactivity are positively related to reactivity and negatively related to predictivity. It also predicts that reward and punishment reactivity are negatively related to each other. Figure 1 shows how, according to PARCS theory, punishment reactivity and reward reactivity relate to each other, and how both of these temperamental tendencies relate to predictivity. In the next paragraphs, we will further argue how this framework may help to elucidate the relation between sensitivity to weak stimuli and overreactivity (punishment reactivity) to strong stimuli.
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| 99.8 |
Schematic overview of the reactive (approach and avoidance) and predictive systems according to the Predictive and Reactive Control Systems (PARCS) theory. Within the boxes, which represent the systems, we provide a description of characteristic information processing/behavior mediated by the given system. The encircled terms indicate the temperamental tendencies that arise from bias toward the given system. The arrows indicate inhibitory relationships between the reactive systems and predictive systems, and between the reactive approach system and reactive avoidance system.
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| 99.8 |
The present empirical paper does not provide the space to review all evidence behind PARCS theory (for this we refer to review papers, e.g., Tops et al., 2010, 2014, 2016), however, we provide some examples from neuroimaging research that show integrated responding to both aversive and appetitive stimuli in areas that also facilitate processing of weak or ambiguous stimuli. Cortical components of the reactive system in PARCS (e.g., the AI and dACC) match what has been named the “salience network” in human neuroimaging studies: a key network in sensory perception and attention allocation (Seeley et al., 2007). Besides receiving projections from networks that seem more strongly involved in either reward or punishment processing, the salience network responds to salient stimuli in general, both appetitive and aversive (Hayes and Northoff, 2012; Hayes et al., 2014). Moreover, higher connectivity within the salience network was found to be associated with higher individual differences scores of harm avoidance and anxiety (Markett et al., 2013) and decreased connectivity of this network with areas of the reward system was related to decreased extraversion in depressed patients (van Tol et al., 2013). At the same time, the salience network seems involved in processing stimulus salience or relevance to a current task (e.g., detecting a sound) and to be activated whenever sensory input poses a challenge by sensory uncertainty or ambiguity, the disambiguation of which requires enhanced effort and alertness (Sterzer and Kleinschmidt, 2010; Lamichhane and Dhamala, 2015).
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| 99.9 |
The evidence from neuroimaging studies for involvement of the salience network in processing of aversive and appetitive stimuli as well as weak or ambiguous stimuli converges with temperament and personality research. In terms of personality, PARCS theory suggests that trait absorption reflects individual inclinations toward salience network activation (Tops et al., 2016). Absorption is defined as the tendency to get attentionally immersed in and elaborately appraise salient sensory or emotional (positive and negative) experiences and one’s internal state (Gohm and Clore, 2000; Tops et al., 2016) and as such corresponds to the attentional functions of the reactive system, which include orienting responses and appraisal of salient stimuli. The notion that absorption is a correlate of the salience network in the reactive system is supported by findings that activation of areas in the salience network showed correlation with trait absorption (Tops and Boksem, 2010) and state absorption (Wilson-Mendenhall et al., 2013; Hsu et al., 2014). It is also in line with findings that participants scoring high on absorption showed enhanced processing of emotionally neutral task relevant stimulus features as well as enhanced processing of task irrelevant emotional features compared to participants scoring low on absorption. This was reflected in reaction times (RTs) as well as in event related brain potentials (ERPs) to a task in which participants determined whether the letter A, the task-relevant stimulus feature, was present in a word or not. Participants scoring high on absorption responded faster to words with the letter A than without the letter A, while low absorption participants did not show such RT difference. High absorption participants also showed an increased sustained widespread positivity to words containing the letter A, labeled as Late Positive Complex (LPC), compared to low absorption participants, indicating enhanced processing of task relevant features. Furthermore, for high absorption participants only, RT was further decreased and the LPC was further increased when the A occurred in a word with emotional compared to emotionally neutral meaning, indicating that processing of task-irrelevant emotional features was also enhanced for these participants (de Ruiter et al., 2003, 2006). Absorption can be measured on various scales (Tops et al., 2016) including the Openness to Experience subscale of personality inventories based on the five factor model of personality (McCrae and Costa, 1987, 1997; Costa and McCrae, 1992). This is supported by studies finding large correlations between openness to experience (especially its fantasy, aesthetics and feelings facets) and other absorption scales, such as the Tellegen Absorption Scale (Glisky et al., 1991; McCrae, 1993). Absorption and openness to experience conceptually also strongly overlap with orienting sensitivity in Evans and Rothbart (2007) temperament model, which is supported by large correlations found between openness to experience and ATQ orienting sensitivity (Wiltink et al., 2006). In the current study we will therefore use ATQ orienting sensitivity as measure of perceptual and attentional aspects of the reactive system. In comparison to other measures of absorption, the ATQ orienting sensitivity scale is particularly suitable for the present study, because its subscales uniquely focus on orienting to and appraising of weak and subtle stimuli (Evans and Rothbart, 2008). This scale thus provides a measure of sensitivity to weak stimuli that reflects perceptual and attentional aspects of reactivity in PARCS theory.
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| 89.94 |
As reviewed above, both the mutually anticorrelated approach (reward) and avoidance (punishment) systems input to, and activate, the cortical areas of the reactive control system. In turn, the activation of reactive control increases the allocation of attentional resources to aversive and appetitive stimuli, as well as to relevant weak stimuli, which is supported by findings on the perceptual and attentional correlates of trait absorption (de Ruiter et al., 2003, 2006) and the perceptual and attentional correlates of salience network activation (Sterzer and Kleinschmidt, 2010; Hayes and Northoff, 2012; Hayes et al., 2014; Lamichhane and Dhamala, 2015). We therefore expect that orienting sensitivity, as measure of perceptual and attentional aspects of reactivity, positively associates to other measures of sensitivity to weak stimuli, such as sensory detection thresholds. We also expect that orienting sensitivity is related to both the mutually anticorrelated traits of discomfort (punishment reactivity to strong stimuli) and high intensity pleasure (reward reactivity to strong stimuli). In the next paragraph, we will argue more specifically, based on PARCS theory, how taking into account both punishment reactivity and reward reactivity may help to understand the relation between sensitivity to weak stimuli and overreactivity (punishment reactivity) to strong stimuli.
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| 99.9 |
As argued above, based on PARCS theory we predict that perceptual sensitivity is related to both punishment and reward reactivity. However, because the two types of reactivity are also negatively related to each other, they are possible suppressor variables that may cancel out the separate positive relations between each reactivity measure and perceptual sensitivity (MacKinnon et al., 2000). We will illustrate the idea of (statistical) suppression with an example in the auditory domain because the current study included a measure of sensitivity in the auditory domain. PARCS theory predicts that sensitivity to weak sounds, given its association with reactivity, is high in individuals who have a tendency to experience aversion (high punishment reactivity) when exposed to noise or loud sounds and also in individuals who have a tendency to experience pleasure (high reward reactivity) when exposed to noise or loud sounds. At the same time, the tendency to experience aversion due to noise or loud sound is inversely related to the tendency to derive pleasure from it: the more one tends to experience aversion from intense sound the less one tends to experience pleasure from it. Now, take an individual who has a very low tendency to experience aversion from strong sound. On the one hand this person is expected to score relatively low on sensitivity to weak sounds. On the other hand, however, this person is also likely to experience strong pleasure from loud sound and therefore is actually likely to score high on sensitivity. If this is the case, extreme scores on both ends of a scale measuring the tendency to experience aversion to strong sounds are associated with high sensitivity. Therefore, across individuals, no positive relationship between deriving displeasure and sensitivity will be observed. This example illustrates that the positive relation between punishment reactivity and perceptual sensitivity can be canceled out due to the negative relation between punishment reactivity and reward reactivity (and vice versa). This type of statistical relationship is known as a suppression effect or inconsistent mediation (MacKinnon et al., 2000). If these suppression effects occur, both types of reactivity will show to be related to sensitivity when controlled for each other, but this relationship is canceled or dampened when not controlled for each other. Thus, according to PARCS theory, in order to gain proper understanding of the relation between sensitivity to weak stimuli and reactivity to strong stimuli, it is crucial to take the suppression effects into account. In the current study we tested the predicted suppression effects.
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| 99.94 |
As discussed above, in the current study we used (sub)scales of the ATQ to measure punishment and reward reactivity to strong stimuli and sensitivity to weak stimuli, thereby building on the work of Evans and Rothbart (2007, 2008). To summarize, the ATQ discomfort scale measured punishment reactivity to strong stimuli, the ATQ high intensity pleasure (HIP) scale measured reward reactivity to strong stimuli, and the ATQ orienting sensitivity scale measured sensitivity to weak stimuli. Table 1 provides definitions and sample items of these ATQ scales. To test the predicted suppression effects we performed a correlational analysis on the ATQ scores. We expected to find a relation between ATQ discomfort and ATQ orienting sensitivity when ATQ HIP would be added as control variable to the analysis, but a weaker, or absent, relation when ATQ HIP would not be taken into account. Similarly, we expected to find a relation between ATQ HIP and ATQ orienting sensitivity when ATQ discomfort would be added as control variable to the analysis, but a weaker or absent relationship when ATQ discomfort would not be taken into account.
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| 100.0 |
Further, building on previous studies investigating the relationship between sensitivity to weak stimuli and traits associated with overreactivity to strong stimuli (Smith, 1968; Siddle et al., 1969; Edman et al., 1979; Ellermeier et al., 2001) we included a psychophysical measure of perceptual sensitivity in addition to ATQ orienting sensitivity. We chose a measure in the auditory domain because previous studies in this domain yielded mixed conclusions regarding the question whether sensitivity and reactivity arise from the same trait. The inclusion of an objective psychophysical measure is also important because ATQ orienting sensitivity is a rating scale measure of sensitivity to weak stimuli. Rating scale measures may be prone to response bias, which is a “systematic tendency to respond to questionnaire items on some basis other than the specific item content” (Paulhus, 1991, p. 17). There are various types of response biases including social desirability bias, where participants answer in such a way that they give a good impression of themselves regardless of their true characteristics (Furnham and Henderson, 1982; Paulhus, 1991), and extremity bias which is the tendency to give extreme rather than moderate responses (or vice versa) irrespective of the content of the items (Bachman and O’Malley, 1984; Paulhus, 1991; Mõttus et al., 2012). Response bias can impact the magnitude of the means and standard deviations of single scales as well as correlations between scales (Baumgartner and Steenkamp, 2001; Van Vaerenbergh and Thomas, 2013). Including an objective psychophysical measure of sensitivity enabled us to check whether our results could be explained by response bias or not. We used the masked auditory detection threshold for pure tones, which reflects listeners’ ability to detect faint sounds in noise, as an objective indicator of sensitivity. To measure it we used a two-interval forced choice (2IFC) procedure combined with a staircase procedure (García-Pérez, 1998). This procedure is regarded as criterion-free, that is, it measures sensitivity irrespective of the response criterion used by the observer (Green and Swets, 1966; Kingdom and Prins, 2010) and thus has minimized vulnerability to effects of response bias. If self-reported sensitivity truly reflects perceptual sensitivity, it should correlate similarly to the reactivity scales as objectively measured sensitivity does. Figure 2 shows an overview of the relationships we aimed to test in the current study.
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| 99.94 |
Relationships as predicted by Predictive and Reactive Control Systems (PARCS) theory and tested in the current study (A) between discomfort (punishment reactivity), high intensity pleasure (reward reactivity), and orienting sensitivity, and (B) between discomfort, high intensity pleasure, and auditory threshold.
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| 99.94 |
Eighty-one participants (Age: M = 20.5, SD = 2.0, 18–27 years; 20 males) with no self-reported hearing problems or depression took part either for course credit or payment. Data from three participants were not included in the analyses because they had strongly deviating thresholds (above the three inter quartile range criterion). The present study was part of a larger investigation with the same participants on affective and temperamental influences on the masked auditory threshold. Part of this investigation contributed to a study on modulation of the masked auditory threshold by mood state (Bolders et al., 2017). In the current study only the threshold measured prior to the mood induction was used to examine its relationships with individual differences in temperament.
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| 100.0 |
Sound was binaurally presented through insert earphones (Etymotic ER-4B microPro). These earphones provide 35 dB external noise attenuation. Stimulus presentation was controlled by E-prime 2 (Schneider et al., 2002) using a computer with a CRT screen (75 Hz refresh rate, 1024 × 768 resolution). Responses were made on a QWERTY keyboard and by using a mouse.
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| 99.6 |
Temperament was assessed using the short version of the ATQ translated into Dutch by Hartman and Majdandžić (2001). This questionnaire consists of 77 items and contains the same constructs and sub constructs as the original ATQ (Evans and Rothbart, 2007). Each item is formulated as a statement. Participants in the current study were asked to indicate how applicable this statement was to them by clicking on the appropriate answer option presented on the computer screen. We used a 6-point scale varying from “Not at all” (1) to “Completely” (6). There was also a “not applicable option,” which was treated as missing data point. Supplementary Table 3 shows the number of items and Cronbach’s alpha per (sub)scale.
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| 100.0 |
Two wav files were created with Audacity software, one to serve as a signal and one as non-signal in the threshold task. The signal was a 1 kHz pure tone, 500 ms in duration with 10 ms ramped on- and offset. An empty sound file of 500 ms served as non-signal. A white noise (20–10 kHz band-filtered) generated with Goldwave software was used as masking noise that was constantly present during the threshold task. Sound levels at output were calculated from the voltages delivered at the earphone input measured with an oscilloscope (Tektronix TDS2002) and the earphone efficiency reported by the earphone manufacturer (108 dB SPL for 1Vrms in a Zwislocki coupler, ER-4 datasheet, Etymotic Research, 1992). The white noise was presented with a voltage delivered at the earphone input that would equal 38 dB SPL output for a 1 kHz tone. Digital sound properties for all sounds were standardized (44 kHz, 16 bit, mono).
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| 99.94 |
An adaptive two interval forced (2IFC) choice task was employed to measure the auditory threshold. At the beginning of each trial a fixation cross was presented in the center of the screen for 1000 ms. This was followed by two observation-intervals which were marked with a number (1 or 2) presented in the center of the screen. The intervals were separated by a blank interval during which only a fixation cross was presented. The three intervals were each 700 ms in duration. On every trial one of the two observation-intervals was pseudo-randomly selected to contain the signal with the constraint that no more than four trials with the same selected interval could occur in succession. The 500 ms signal was centered in the 700 ms observation interval. After the second observation interval there was a 100 ms blank screen. This was followed by a red “X” presented in the center of the screen until the participants indicated whether they had heard the signal in the first or the second interval by pressing the z-key on the keyboard with their left index finger or the m-key on the keyboard with their right index finger, respectively. The sound level of the signals depended on the performance of the participants and increased or decreased adaptively according to a transformed and weighted up/down rule (García-Pérez, 1998). A 1-up/2-down rule was used and the ratio of the step size down and step size up was 0.548. In other words, after one incorrect trial the sound level of the tone went up one step (e.g., 3 dB), but it went down one step only after two consecutive correct trials, with the step size up being 1.82 times the size of the step down. This rule has been shown to reliably converge to 80.35% correct performance (García-Pérez, 1998). Initially the step size down was 15 dB. This changed to 5 dB after two reversal points (trials at which the sound level changed from going up to down or vice versa) and to 3 dB after four more reversal points. The initial sound level was 68 dB SPL. To calculate the threshold (sound level needed for a performance of 80.35% correct) the sound levels of tones at the last ten reversal points were averaged. The e-prime script for the adaptive procedure was adapted from Hairston and Maldjian (2009).
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| 100.0 |
After providing informed consent, participants were seated in a comfortable chair at 50 cm from the computer monitor in a quiet dimly lit individual test cubicle. They were instructed about the flow of the experiment, practiced with correct earphone insertion and the experimenter verified whether external sounds were indeed attenuated. Further instructions followed on the computer screen. Regarding the threshold task it was explained that the signal would be presented equally often in each interval, and that, although the signal could be difficult to hear on some trials, it was important to keep paying attention to the task and that an answer was required on all trials. The task instructions stressed accuracy and all responses were self-paced. Participants carried out eight practice trials in order to get used to the task. The practice trials were equal to the trials of the threshold task, except that the sound level of the signals was kept at 68 dB SPL and after each practice trial participants received feedback about their accuracy. Following the practice trials the threshold task started. At the end of the study participants filled out the ATQ.
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| 100.0 |
To validate the self-report measure of sensitivity, we correlated the threshold with ATQ orienting sensitivity. ATQ orienting sensitivity includes perceptual sensitivity as a subscale (five items), but also includes subscales measuring sensitivity to experiencing divergent mental associations or images (five items) and sensitivity to subtle affective stimuli (five items). Because ATQ orienting sensitivity is a more reliable (cronbach’s α = 0.67 in current study) measure of sensitivity to subtle stimuli than each of its subscales separately (see Supplementary Table 3 for cronbach’s αs), we used the orienting sensitivity scale in subsequent analyses as self-report measure of sensitivity. It is worth mentioning here that the threshold correlated similarly with each ATQ orienting sensitivity subscale (see Supplementary Table 2 for correlation coefficients).
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| 100.0 |
To answer our main question about the relation between sensitivity to weak stimuli and reactivity to strong stimuli, we examined the partial correlations between ATQ orienting sensitivity and the perceptual reactivity scales (ATQ discomfort and ATQ HIP). The correlation between ATQ orienting sensitivity and ATQ discomfort was controlled for ATQ HIP and the correlation between ATQ orienting sensitivity and ATQ HIP was controlled for ATQ discomfort. We carried out the same analyses replacing ATQ orienting sensitivity with the masked auditory threshold.
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| 100.0 |
To examine the expected suppression effects we repeated the above described correlational analyses but without controlling for ATQ HIP or ATQ discomfort. If ATQ HIP and ATQ discomfort suppress each other’s association with perceptual threshold and sensitivity, then not controlling for the suppressing variable should substantially reduce the correlations.
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| 100.0 |
In all of the main analyses several other variables were controlled for. First, because we were interested in the relationships with the perceptual aspects of punishment and reward reactivity, we controlled for the broader constructs of ATQ frustration and ATQ positive affect. Where ATQ discomfort measures irritability due to intense stimulation (e.g., “I find loud noises to be very irritating”), ATQ frustration measures irritability in general (e.g., “It doesn’t take very much to make me feel frustrated or irritated”). And, where ATQ HIP measures the tendency to experience pleasure due to intense stimuli (e.g., “I would enjoy the sensation of listening to loud music with a laser light show”), ATQ Positive Affect measures the tendency to experience pleasure in general (e.g., “It doesn’t take much to evoke a happy response in me”). We controlled for ATQ frustration because people who indicate on the ATQ discomfort scale that they get irritated from intense stimuli may actually get irritated easily in general, not only by intense stimuli. Similarly, we controlled for ATQ positive affect because people who indicate on the ATQ HIP scale that they derive pleasure from intense stimuli may actually tend to derive pleasure from things more in general, not specifically from intense perceptual stimuli.
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| 100.0 |
Second, we also controlled for sex because this variable has been associated with discomfort and HIP (Ormel et al., 2005) and hearing sensitivity (Robinson, 1988). Age has also been associated with discomfort or unpleasantness experienced due to high arousal stimuli (Keil and Freund, 2009; Tops and Matsumoto, 2011) and with hearing sensitivity (Robinson, 1988). However, the age range was small and adding age as a control variable did not affect the pattern of the correlations. Therefore the analyses are presented without controlling for this variable.
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| 100.0 |
To validate the self-report measure of sensitivity, we correlated the threshold (M = 21.14 dB SPL, SD = 1.93) with ATQ orienting sensitivity. The threshold had a moderate negative relationship with orienting sensitivity, r = -0.31, p = 0.006. The Supplementary Material provides a full matrix of the uncorrected correlations between all ATQ scales (Supplementary Table 1) and subscales (Supplementary Table 2) and the threshold. It also provides a table with descriptive statistics for the ATQ (sub)scales (Supplementary Table 3).
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| 100.0 |
Correlation coefficients and significance levels of the relationships tested for the main analyses are provided in Figure 3. ATQ orienting sensitivity displayed significant partial correlations with ATQ discomfort and trends of similar magnitude with ATQ HIP when ATQ discomfort and ATQ HIP were controlled for each other and for ATQ frustration, ATQ positive affect, and sex. Similarly, the masked auditory threshold displayed significant partial correlations with ATQ discomfort and with ATQ HIP, when ATQ discomfort and ATQ HIP were controlled for each other and for ATQ frustration, ATQ positive affect, and sex.
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| 100.0 |
Overview of the partial correlation coefficients and significance levels of the relationships (A) between discomfort (punishment reactivity), high intensity pleasure (reward reactivity), and orienting sensitivity and (B) between discomfort, high intensity pleasure, and auditory threshold. 1Controlled for sex, frustration, positive affect, and HIP or discomfort. 2Controlled for sex, frustration, positive affect. †p < 0.10, ∗p < 0.05, ∗∗p < 0.01, ∗∗∗p < 0.001.
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| 99.94 |
When ATQ discomfort and ATQ HIP were not controlled for each other, most correlations between the (self-reported and objective) sensitivity and self-reported reactivity measures were low and not significant. Only ATQ orienting sensitivity still correlated with ATQ discomfort, albeit with slightly lower magnitude.
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| 100.0 |
The present study examined the relations between self-reported and objectively measured sensitivity to subtle stimuli and self-reported reactivity to strong stimuli within the framework of PARCS theory. Importantly, two types of reactivity are distinguished in this theory: punishment reactivity and reward reactivity. We measured punishment reactivity to strong stimulation by means of the ATQ discomfort scale and reward reactivity to strong stimulation with the ATQ HIP scale. Sensitivity to weak stimuli was measured using the objectively determined masked auditory threshold as well as the ATQ orienting sensitivity scale, which is a self-report measure of sensitivity to weak stimuli that reflects perceptual and attentional aspects of reactivity in PARCS theory.
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study
| 100.0 |
As predicted from PARCS theory, our results showed that ATQ orienting sensitivity was positively associated with objectively determined sensitivity (inverse of the masked auditory threshold). Furthermore, and crucial for answering our research question, both types of reactivity to strong stimulation related to (self-reported and objectively measured) sensitivity to weak stimuli, but only when controlled for each other, indicating a mutual suppression effect. These findings are in line with the notion of PARCS theory that punishment and reward reactivity overlap in terms of reactivity toward stimulation, but that these tendencies also oppose each other in terms of response orientation (approach or avoid). Note that associations between the reactivity measures and self-reported sensitivity to weak stimuli were replicated when sensitivity was objectively measured as the masked auditory threshold, which makes it unlikely that the associations between sensitivity to weak stimuli and the reactivity to strong stimuli were driven by response bias.
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study
| 100.0 |
Our results may help to understand previous inconsistencies in the literature with respect to the dependency of overreactivity to strong stimuli and perceptual sensitivity to weak stimuli. Some studies found support for the claim that sensitivity to weak stimuli and overreactivity arise from the same trait (Smith, 1968; Siddle et al., 1969; Edman et al., 1979), while other research findings suggested that sensitivity to weak stimuli and overreactivity are independent traits (Ellermeier et al., 2001; Evans and Rothbart, 2008). Our study demonstrated that when not controlled for reward reactivity to strong stimuli, the relation between punishment reactivity to strong stimuli and perceptual sensitivity might be suppressed. If this is the case, the dependency between punishment reactivity and sensitivity becomes apparent only when reward reactivity is kept constant. Discrepant and zero findings can be expected because differences in the distributions of reward reactivity introduce differences in the relation between punishment reactivity to strong stimuli and sensitivity to weak stimuli. The same holds for the relation between reward reactivity to strong stimuli and sensitivity to weak stimuli when not controlled for punishment reactivity. As previous studies did not take both punishment and reward reactivity to strong stimuli into account, this might explain their discrepant findings regarding the dependency of overreactivity to strong stimuli and perceptual sensitivity to weak stimuli.
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study
| 99.94 |
Because Eysenck’s (1967) personality theory is such a well-known and influential theory that considers sensitivity to weak stimuli and overreactivity to strong stimuli as resulting from one underlying trait, we will specifically compare our results to this theory. Our findings are in line with the ideas of Eysenck in the sense that overreactivity to strong stimuli and sensitivity to weak stimuli seem to be dependent. The pattern of dependencies we found, however, does not agree with Eysenck’s predictions, which are based on introversion-extraversion as underlying trait. According to Eysenck, overreactivity to high intensity stimulation is associated with high sensitivity to weak stimulation (Eysenck, 1967). This does match our finding that punishment reactivity to strong stimuli, when controlled reward reactivity, was associated with sensitivity to weak stimulation. Eysenck (1967), however, also suggested that extraverts’ enjoyment of high intensity stimulation is associated with low sensitivity to weak stimulation. By contrast, we found that, when controlled for punishment reactivity to strong stimulation, pleasure from high intensity stimulation was associated with high sensitivity to weak stimulation. This renders it unlikely that sensitivity to weak stimuli and overreactivity are associated due to introversion-extraversion as underlying trait. Instead, as suggested by PARCS theory the pattern of dependencies can be explained by individual differences in the tendency to activate the salience network of the reactive system. This network mediates processing of aversive and appetitive stimuli as well as relevant weak stimuli, and receives input from both the mutually anticorrelated approach (reward) and avoidance (punishment) reactive systems. Furthermore, as discussed above, our findings stress the importance of taking into account both reward and punishment reactivity to understand the relationship between sensitivity to weak and overreactivity to strong stimuli. This favors the use of these two dimensions, which are based on Gray’s early conceptions of personality (Gray, 1970, 1989) over Eysenck’s introversion–extraversion dimension in studying the relation between sensitivity and overreactivity. Moreover, Gray’s (1970, 1989) dimensions not only fit better with the current results, but also seem to better account for earlier findings. According to Gray (1970, 1989) punishment reactivity (which he labeled anxiety) is, in terms of Eysenck’s personality dimensions, reflected in a combination of high introversion and high neuroticism. Relating sensitivity to punishment reactivity rather than to introversion thus seems to better account for findings of confounding and interaction effects by neuroticism in earlier studies that investigated the relation between introversion and the threshold for noticing weak stimuli (Siddle et al., 1969; Edman et al., 1979).
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study
| 99.94 |
In addition to furthering the understanding of the relation between sensitivity to weak and reactivity to strong stimuli, our findings also have relevant implications for the study of temperamental and psychophysical determinants of noise annoyance or annoyance produced by other environmental nuisances in daily life. This regards, for example, the question whether noise sensitivity, a measure of discomfort in the auditory domain that predicts noise annoyance, is dependent on basic auditory perception or not (Ellermeier et al., 2001). As our results demonstrated, perceptual reward reactivity (HIP) may suppress the relation between auditory punishment reactivity (discomfort) and perceptual sensitivity. Therefore, in order to gain a more complete picture of the determinants of noise annoyance, we recommend including measures of reward reactivity to strong stimuli in future studies and controlling for it. In addition to noise annoyance, PARCS theory might also provide a framework to contribute to the understanding of other environmental intolerances, such as multiple chemical sensitivity (MCS). MCS, also known as idiopathic environmental intolerance (IEI), is a condition that is characterized by intolerance for chemical agents expressed as various somatic complaints including fatigue, headaches and pain (Graveling et al., 1999; Bornschein et al., 2002). Interestingly, MCS has been associated with anxiety and harm avoidance (Hillert et al., 2013) as well as with absorption (Witthöft et al., 2008) which are all indices of reactivity in PARCS theory.
|
study
| 99.7 |
The present study does have some limitations. First, our study had a relatively small number of participants (N = 78). Second, although the ATQ provided suitable measures that have been used in related research before (Evans and Rothbart, 2008), no scales have yet been developed specifically to measure the constructs derived from PARCS theory. Third, our objective measure of sensitivity concerned auditory sensitivity only. Replication of our findings in a large sample using objective measures in other sensory modalities and using a questionnaire based on PARCS theory is important to show robustness and generalizability of these findings. In addition, future studies may also benefit from psychophysical measures of the response to strong stimuli, such as the threshold of pain and objective measures of reward and punishment reactivity such as ERP responses to reward and punishment stimuli during task performance (e.g., Boksem et al., 2006, 2008).
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study
| 100.0 |
Taken together, our study showed that self-reported as well as objectively assessed sensitivity to weak stimuli was associated with punishment and reward reactivity to strong stimuli. These relationships only became apparent when the reactivity measures were controlled for each other, indicating a mutual suppression effect. The fact that previous studies did not take this suppression effect into account may explain previous discrepant findings concerning the relation between sensory sensitivity and overreactivity. To conclude, our study indicates that sensitivity to weak stimuli overlaps, at least partly, with the tendency for overreactivity to strong stimuli, in a manner that is in line with the predictions of PARCS theory.
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study
| 100.0 |
All authors were involved in designing the study. AB performed the research. MT and AB analyzed the data. AB drafted the manuscript with input from GB, MT, and PS. All authors critically revised the manuscript and approved the final version for submission.
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other
| 99.94 |
The relationship between energetic gain and locomotory cost is a key determinant in predatory behaviour and greatly influences predator–prey interactions (e.g. Sinclair et al., 2003; Scharf et al., 2006). In the broadest sense, predatory behaviour of mammalian carnivores spans a range from ambushes [e.g. lions (Panthera leo) and leopards (Panthera pardus)] to rapid, long-distance pursuits [e.g. cheetah (Acinonyx jubatus) and spotted hyena (Crocuta crocuta); e.g. Bro-Jørgensen, 2013]. A particularly intriguing case involves the interactions of polar bears (Ursus maritimus) and lesser snow geese (Chen caerulescens caerulescens), a land-based prey that may become an increasingly important seasonal food resource for polar bears as climate changes (Gormezano and Rockwell, 2013a,b, 2015).
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review
| 99.7 |
Polar bears normally use the sea ice as a platform to catch marine prey, particularly ringed seals (Pusa hispida), and accumulate a majority of their annual fat reserves from consuming seal pups in spring (e.g. Stirling and Øritsland, 1995). In more southern polar bear populations, it is thought that this energy store helps to sustain the bears during the ice-free period each summer (e.g. Stirling and Derocher, 1993; Regehr et al., 2007). With warmer temperatures leading to earlier sea ice break-up, access to this energy-rich spring seal diet may become limited, potentially forcing the bears to expend energy seeking land-based food to compensate for energy deficits (e.g. Stirling and Derocher, 2012; Gormezano and Rockwell, 2013a, 2015; Lunn et al., 2016). Any increased effort to obtain food is of concern because polar bears are considered inefficient at walking (Øritsland et al., 1976; Best 1982; Hurst et al., 1982a,b), exhibiting higher rates of oxygen consumption with increased walking speed than predicted for mammals of their size (Taylor et al., 1970; Fedak and Seeherman, 1979). The higher rates of energy use have been attributed to their morphology, particularly their large, heavy limbs (Øritsland et al., 1976; Hurst et al., 1982a,b), a characteristic shared by male lions that likewise have relatively high costs of locomotion (Chassin et al., 1976). Despite these energetic limitations, polar bears are known to walk long distances in search of prey on sea ice and land (e.g. Born et al., 1997; Amstrup et al., 2000; Parks et al., 2006; Anderson et al., 2008; Rockwell et al., 2011) but generally use more energy-conserving stalking or ‘still-hunting’ techniques to capture seals and other marine mammals on the sea ice (e.g. Stirling, 1974; Smith, 1980).
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study
| 99.8 |
Some polar bears, especially those forced ashore when the sea ice melts in summer, have been observed running on land in pursuit of terrestrial prey (e.g. Brook and Richardson, 2002; Iles et al., 2013 and references therein). Given their locomotive inefficiency and potential to overheat in warm weather (Øritsland, 1970; Øritsland and Lavigne, 1976; Best, 1982), it is unclear whether these more intensive pursuits can be energetically profitable (Lunn and Stirling, 1985; Iles et al., 2013). In the only examination of this issue thus far, Lunn and Stirling (1985) used a calculation based on Hurst et al. (1982a) to suggest that a 320 kg polar bear chasing a goose at 20 km/h for >12 s would expend more energy in the pursuit than could be obtained from consuming it. Despite the speed and mass specificity of that projection, many authors have used this threshold in evaluating observations of polar bears chasing various land-based prey [e.g. caribou, Rangifer tarandus (Brook and Richardson, 2002); barnacle geese, Branta leucopsis (Stempniewicz, 2006); thick-billed murres, Uria lomvia (Donaldson et al., 1995); lesser snow geese (Iles et al., 2013)] and questioned the energetic worth of the observed predatory behaviours.
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study
| 60.84 |
The exact energetic costs associated with land-based hunting behaviour are especially important for polar bears in western Hudson Bay, where recent warming trends are rapidly diminishing ice extent and duration (Gagnon and Gough, 2005; Stirling and Parkinson, 2006; Lunn et al., 2016). If polar bears come ashore with nutritional deficits (e.g. Stirling and Parkinson, 2006; Regehr et al., 2007), any calories obtained on land may become increasingly important for survival (Gormezano and Rockwell, 2013a,b; Gormezano, 2014; Gormezano and Rockwell, 2015) unless the net energetic gain from foods obtained on land exceeds the energetic costs required to obtain them. In western Hudson Bay, snow geese make up an increasing proportion of polar bears’ land-based diet owing in part both to increased temporal overlap of the two species and to greatly increased abundance of snow geese (Gormezano and Rockwell, 2013a, 2015). Given that polar bears in this region spend increasingly more time on land and thus have more opportunities for terrestrial foraging, we constructed predictive models that estimate, for the first time, the metabolic costs of terrestrial locomotion for polar bears of multiple sizes travelling at various speeds. We then use the best-fitting model to evaluate when a polar bear would profit from chasing and catching moulting snow geese, a common terrestrial prey species during summer.
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study
| 100.0 |
In the following analysis, we revisit the only published data on the metabolic costs of locomotion across a range of speeds for polar bears of multiple sizes. We assess the profitability of pursuing flightless geese using data-driven energetic models that simultaneously account for the effects of polar bear speed and mass. We show that pursuits lasting longer than 20 min in duration can be energetically profitable, although this depends importantly on the speed and mass of polar bears, and that successful pursuits of even distant geese can result in net energetic gains for some polar bears. Furthermore, we show that the smaller-sized and younger bears that could take more advantage of this profitability include those whose survival in western Hudson Bay is lower (Lunn et al., 2016) and that may be more impacted by climate change (Regehr et al., 2007).
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study
| 100.0 |
To develop a data-driven model that allows oxygen consumption (and thus metabolism) to scale with polar bear speed and mass, we extracted original data from the three published studies that reported measurements of oxygen consumption (V˙O2; in millilitres of O2 per gram per hour) as a function of walking speed for polar bears that weighed 125, 155, 190 and 235 kg. The 125 and 155 kg animals were subadult males (as defined by Watts et al., 1991), the 190 kg animal was a 4-year-old female (Hurst et al., 1982a) and the 235 kg animal was a ~4-year-old male (Øritsland et al., 1976). We used the means of the multiple trials of each bear at each speed as the best estimates of O2 consumption for each mass and speed. Both linear (Øritsland, 1970; Hurst et al., 1982a) and double exponential regression models (Hurst et al., 1982a) have previously been used to describe how oxygen consumption changes with speed for bears of different sizes. Here, we first considered three potential models to describe the general shape of the relationship between polar bear speed [S; we use this term rather than velocity (V) as used by Hurst et al., 1982a] and oxygen consumption (V˙O2) using data from Øritsland et al. (1976), Hurst et al. (1982a) and Watts et al. (1991). Our initial model set included the following:
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study
| 100.0 |
and (3) a double-exponential model that allows metabolism to more flexibly scale with speed, (3)V˙O2=PebSc; where P is polar bear postural cost (i.e. the energetic cost of maintaining an upright posture when speed is zero), e is the natural log (2.718…), and b and c are exponents that describe the rates at which oxygen consumption changes with movement speed (S). From previous work (Hurst et al., 1982b), postural costs are known to depend on mass. Thus, in all models we fixed the postural costs at the expected values for each polar bear mass based on the equation of Hurst et al. (1982b), following Taylor et al. (1970): (4)P=1.056×mass−0.25.
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study
| 100.0 |
By fixing the postural costs (the y-intercept) based on this equation rather than allowing the postural costs to be estimated based on model fit, we improve the biological realism of our models outside the range of our data (i.e. when speed is zero), while only slightly sacrificing goodness of fit within the range of our data (speeds of 1.8–7.92 km/h). We note, however, that results were qualitatively similar whether postural costs were fixed based on Equation 4 or estimated based on our data. We evaluated relative support for the models using Akaike's information criterion (AICc; Akaike, 1973) and found that the exponential and double-exponential models received similar support (Table 1; ΔAICc = 0 and 0.5, respectively), and greatly outperformed the linear model (ΔAICc = 24). Table 1:Model selection results incorporating effects of mass on the relationship between speed and oxygen consumption.ModellogLikAICcΔLogLikΔAICcparametersWeightPebS 10.1−15.512020.288PebSc 11.3−1513.20.530.223Pe(b+m1*mass)S(c+m2*mass) 14.9−14.716.70.750.199PebS(c+m2*mass) 12.3−13.614.21.940.113Pe(b+m1*mass)Sc 12.2−13.414.12.140.101Pe(b+m1*mass)S 10.3−12.812.12.730.076P+bS −1.88.50242<0.001Model parameters are as follows: b and c, single and double exponents, respectively; e, the natural logarithm (2.718…); m1 and m2, scaling parameters that relate the single exponent and the double exponent, respectively, to polar bear mass; mass, polar bear mass (in kilograms); P, postural costs; and S, polar bear movement speed. In all models, postural costs are described by Equation 4 and thus depend on polar bear mass.
|
study
| 100.0 |
Model parameters are as follows: b and c, single and double exponents, respectively; e, the natural logarithm (2.718…); m1 and m2, scaling parameters that relate the single exponent and the double exponent, respectively, to polar bear mass; mass, polar bear mass (in kilograms); P, postural costs; and S, polar bear movement speed. In all models, postural costs are described by Equation 4 and thus depend on polar bear mass.
|
study
| 99.94 |
We then constructed several additional models to evaluate potential effects of polar bear mass on oxygen consumption, beyond the effects on postural cost in Equation 4. Given that the exponential and double-exponential models received similar support and produced similar predictions across the range of our data, we constructed a suite of models that allowed mass to influence b and/or c in Equations 2 and 3 (Table 1). We used AICc and Akaike weights to evaluate relative support among different parameterizations and assess the relative effects of mass and speed on oxygen consumption.
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study
| 100.0 |
Using model projections of oxygen consumption based on our top model and following Lunn and Stirling (1985), we calculated the time threshold (hereafter, ‘inefficiency threshold’) beyond which the calories expended to chase a goose exceeded the calories obtained from consuming it for polar bears ranging in mass from 125 to 235 kg and over a range of speeds from 0 to 7.9 km/h. For comparative purposes with previous work (Lunn and Stirling, 1985) and because polar bears are known to run at speeds up to 29 km/h (Harrington, 1965), we also projected inefficiency thresholds to 20 km/h. We discuss the assumptions and limitations of those extrapolations in the Discussion.
|
study
| 100.0 |
Estimating the usable energy available to a polar bear eating a goose requires knowledge of (i) the energy in the part(s) of a goose that are eaten, and (ii) the digestibility of the energy in the parts of the goose eaten. Polar bears that successfully capture and eat a variety of prey including seals (Smith, 1980; Best, 1985) and geese (Iles et al., 2013, Gormezano and Rockwell, 2015; DTI & RFR personal observations) rarely consume the less digestible portions, including hair and feathers, and usually avoid eating the gastrointestinal tract and the entire skeleton. Thus, we assumed that polar bears primarily consumed the breast, leg muscle, gizzard and fat stores from a captured goose. We estimated the caloric value of these eaten parts of the goose using adult female goose body composition data from Ankney and MacInnes (1978) (as did Lunn and Stirling, 1985) during the post-hatch period, when many instances of predation have been observed (Iles et al., 2013). At this post-hatch time, adult female geese (n = 35) had negligible amounts of fat and 163.3 ± 4.0 g of protein within the gizzard, breast and leg muscles (Table 3 of Ankney and MacInnes, 1978), which would provide 702.5 kcal, assuming an energy-to-protein conversion of 4.3 kcal/g protein (Robbins, 1993). However, polar bears cannot be expected to digest all the available protein, so some discount is necessary.
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study
| 100.0 |
Grizzly and black bears digested 89–96% of crude protein in the meat from various mammals and birds (Pritchard and Robbins, 1990), whereas the digestibility of crude protein for bears fed whole birds or mammals was less (85.5 ± 2.2%) because of the non-digestible or less digestible parts (e.g. feathers, hair, skeleton; Pritchard and Robbins, 1990; Robbins, 1993). Likewise, captive polar bears fed various parts of ringed seals (Phoca hispida) digested 72–95% of protein nitrogen, with the highest digestibility occurring when polar bears ate seal muscle and viscera and the lowest digestibility when the skeleton, skin and blubber were also eaten (Best, 1985). We assumed that polar bears digested 95% of protein when eating only the gizzard, leg and breast muscle of the goose; digestibility of protein would be much lower (72–85%) if polar bears also ingested other less digestible parts of the whole goose. We present results for the most likely scenario, where polar bears ate the gizzard, leg and breast muscle of the goose and thus gained 667.4 kcal per goose (total of 702.5 kcal, of which 95% was digested).
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study
| 100.0 |
Finally, to determine the conditions in which inefficiency thresholds would be reached during pursuits of flightless geese, we calculated the duration of pursuits resulting from different combinations of polar bear speeds and initial distances from geese. We assumed that geese fled from pursuing bears at 2 m/s; a value slightly higher (and thus more conservative in terms of polar bear profitability analysis) than the reported maximal sustained running speeds of 0.8–1.2 m/s, considered ‘moderate’ to ‘fast’ for similar sized geese (Codd et al., 2005; Hawkes et al., 2014). We calculated the time (t) required for a polar bear to capture a goose as follows: (5)t=DSbear−Sgoose, where D is the initial distance between the bear and the goose, and Sbear and Sgoose are their respective speeds. For each combination of bear mass, speed and initial distance, we calculated the inefficiency threshold and compared this with the chase duration to determine whether the pursuit resulted in a net surplus of energy for the bear.
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study
| 100.0 |
The relationship between polar bear movement speed and oxygen consumption was best described by either an exponential or a double-exponential model, indicating that metabolism increases exponentially at higher speeds (Fig. 1). We found no support for an effect of polar bear mass on the exponents in either model (Table 1). Given that postural cost depends on polar bear mass (Equation 4) but the shape of the exponential relationship between polar bear speed and oxygen consumption does not, larger bears are more efficient than smaller bears on a proportional basis across all movement speeds (Fig. 2). As the exponential model received slightly higher support and was more parsimonious (i.e. used fewer parameters) than the double-exponential model, we used the exponential model to generate estimates of oxygen consumption as a function of polar bear mass and speed (Fig. 2) and, subsequently, to determine energetic inefficiency thresholds and profitability while chasing flightless geese. We noted, however, that the double-exponential model produced very similar predictions to the top model across the range of data (Fig. 1, compare continuous and dashed lines). Figure 1:Mass-specific oxygen consumption increases with movement speed. Postural costs (y-intercept) are affected by polar bear mass according to Equation 4. The top model based on AICc was a single-exponential model (continuous lines). A double-exponential model received similar support (ΔAICc = 0.5) and made similar predictions across the range of data (dashed lines). Figure 2:Mass-specific oxygen consumption (V˙O2) increases with movement speed. Postural costs (y-intercept) are affected by polar bear mass according to Equation 4. Larger bears are proportionately more efficient than smaller bears. Curves are based on predictions from the top model (exponential model; Equation 2), which when parameterized is: V˙O2 = (1.056 * mass−0.25) * e0.2626*S.
|
study
| 100.0 |
Mass-specific oxygen consumption increases with movement speed. Postural costs (y-intercept) are affected by polar bear mass according to Equation 4. The top model based on AICc was a single-exponential model (continuous lines). A double-exponential model received similar support (ΔAICc = 0.5) and made similar predictions across the range of data (dashed lines).
|
study
| 99.94 |
Mass-specific oxygen consumption (V˙O2) increases with movement speed. Postural costs (y-intercept) are affected by polar bear mass according to Equation 4. Larger bears are proportionately more efficient than smaller bears. Curves are based on predictions from the top model (exponential model; Equation 2), which when parameterized is: V˙O2 = (1.056 * mass−0.25) * e0.2626*S.
|
study
| 99.9 |
Combining results from our oxygen consumption models with the energetic value of a female lesser snow goose, we calculated that a 125 kg polar bear could chase a goose for 26.9 min at 7.9 km/h (the maximal speed of polar bears for which oxygen consumption measurements were recorded) before it becomes energetically unprofitable. In contrast, the inefficiency threshold for a 235 kg bear at 7.9 km/h was 16.7 min. Given that energy consumption increases with speed, the inefficiency threshold decreases with increasing speed for bears of any mass. Despite larger bears having lower proportional oxygen consumption than smaller bears (Fig. 2), the higher absolute mass of larger bears results in lower inefficiency thresholds across the range of speeds for which there are data (Fig. 3). As a consequence, smaller bears can sustain chases that are longer in duration. Figure 3:Time ‘inefficiency’ threshold, beyond which the calories expended by a polar bear to chase an adult female goose exceed the calories obtained from consuming it, as a function of speed of the chase and polar bear mass. Note that projections for speeds >7.9 km/h (dashed vertical line) are extrapolations beyond the available data and should be interpreted with caution, but are pictured for comparison with extrapolations by previous studies. The inefficiency threshold (I) is calculated as follows: I = 667.4/(V˙O2 * mass * 4.735)/60, where 667.4 is the caloric value of a goose, mass-specific oxygen consumption (V˙O2) is estimated as in the legend to Fig. 2, and 4.735 is the standard conversion of 1 litre of oxygen to kilocalories.
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study
| 100.0 |
Time ‘inefficiency’ threshold, beyond which the calories expended by a polar bear to chase an adult female goose exceed the calories obtained from consuming it, as a function of speed of the chase and polar bear mass. Note that projections for speeds >7.9 km/h (dashed vertical line) are extrapolations beyond the available data and should be interpreted with caution, but are pictured for comparison with extrapolations by previous studies. The inefficiency threshold (I) is calculated as follows: I = 667.4/(V˙O2 * mass * 4.735)/60, where 667.4 is the caloric value of a goose, mass-specific oxygen consumption (V˙O2) is estimated as in the legend to Fig. 2, and 4.735 is the standard conversion of 1 litre of oxygen to kilocalories.
|
study
| 99.94 |
Ultimately, the time required to capture terrestrial prey depends on the initial distance between the polar bear and prey and the relative speeds of the bear and the prey. If the chase duration exceeds the energy inefficiency threshold for that particular pursuit speed, polar bears will lose energy even from pursuits in which they successfully capture geese. We found that polar bears were capable of capturing geese before reaching their inefficiency threshold for a wide range of pursuit scenarios (Fig. 4, blue areas). Smaller bears (i.e. 125 kg) were capable of gaining energy from pursuits of geese up to 754 m away, whereas larger bears (i.e. 235 kg) could gain energy from pursuits of geese up to 468 m away. Figure 4:Profitability of capturing flightless snow geese for polar bears weighing 125 (A) or 235 kg (B). The initial distance from flightless geese and polar bear speed influence the time required to capture a goose, whereas polar bear mass and speed influence the inefficiency threshold (chase duration beyond which energy expenditures exceed energy gains from consuming a 667.4 kcal goose). Chases that are shorter in duration than the inefficiency threshold are coloured in blue (resulting in a net energy surplus for polar bears). Note that because geese are capable of running at 2 m/s (or 7.2 km/h), bears are incapable of capturing geese when moving slower than this speed. Areas to the right of the white dashed lines are extrapolations outside the range of data, but are pictured for comparison with extrapolations in previous studies.
|
study
| 100.0 |
Profitability of capturing flightless snow geese for polar bears weighing 125 (A) or 235 kg (B). The initial distance from flightless geese and polar bear speed influence the time required to capture a goose, whereas polar bear mass and speed influence the inefficiency threshold (chase duration beyond which energy expenditures exceed energy gains from consuming a 667.4 kcal goose). Chases that are shorter in duration than the inefficiency threshold are coloured in blue (resulting in a net energy surplus for polar bears). Note that because geese are capable of running at 2 m/s (or 7.2 km/h), bears are incapable of capturing geese when moving slower than this speed. Areas to the right of the white dashed lines are extrapolations outside the range of data, but are pictured for comparison with extrapolations in previous studies.
|
study
| 97.75 |
The best-supported predictive model for estimating the metabolic costs of terrestrial locomotion for polar bears of different sizes was a simple exponential model (Fig. 2). Importantly, the shape of the exponential relationship between polar bear speed and metabolic cost did not depend on polar bear mass, and only the postural costs (y-intercept) were mass dependent; the implication being that smaller bears therefore spend proportionately more energy for locomotion than larger bears (Fig. 3). Previous studies have shown that postural costs (energy costs when speed is zero) are greater for smaller bears (Scholander et al., 1950; Hurst et al., 1982b), a pattern observed in smaller and immature animals in general (Taylor et al., 1970; Lavigne et al., 1986). These higher postural costs with decreasing polar bear mass combined with similar exponential increases in the energy costs of locomotion with travel speed regardless of mass result in smaller bears having proportionately higher locomotion costs than larger bears at a given travel speed.
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study
| 100.0 |
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