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668d5c5501103d79c580b9ba
| 23 |
To synthesize ACM-labeled PCL 45 -b-PDMA 348 , ACM was coupled to the PCL 45 -b-PDMA 348 polymer backbone by esterification. In a typical coupling reaction, PCL 45 -b-PDMA 348 (200 mg, 1 eq), ACM (4.3 mg, 3 eq), DMAP (0.6 mg, 1 eq) and DCC (10.3 mg, 10 eq) were mixed in an ampule with 2 mL CHCl 3 . The solution was left stirring at room temperature for 2 days. The solution was then filtered, and the filtrate precipitated in diethyl ether three times and the resultant polymer was dried under vacuum.
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| 24 |
SEC analysis was performed on an Agilent 1260 Infinity II system combined with refractive index (RI) and ultraviolet (UV) detectors (λ = 309 and 360 nm), equipped with a PLGel 3 µm (50 × 7.5 mm) guard column and two PLGel 5 µm (300 × 7.5 mm) mixed-D columns, mobile phase (eluent) is CHCl 3 with 0.5% triethylamine (TEA). Molecular weight (M w ) and molecular weight distributions were calibrated against poly(methyl methacrylate) (PMMA) standards and analyzed using Agilent SEC software. without stirring on a heating block for 3 h before cooling to room temperature and then aging for 5 days to yield micron-length polydisperse fibers (Fig. ). The crystalline fibers were then sonicated using a Bandelin Sonopuls sonication probe in a dry-ice/acetone bath for 20 min. The polymer solution was exposed to 60 cycles of 20 s bath sonication with an interval of 100 s to yield a short crystalline seed stock solution (Fig. ). For determination of the average length of the seed, a minimum number of 100 seeds were analyzed with TEM (Fig. ). to form a targeted concentration, then aged for at least 1 h. Then a small volume of unimer stock solution was added to a dispersion of seed solution in a screw cap vial followed by shaking by hand for 5 s, the final unimer concentration was varied by adjusting the volume of stock solutions added to each reaction. At predetermined time points, 140 µL sample mixture was withdrawn and added to 140 µL of deionized water to quench the reaction, followed by subsequent analysis.
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In situ 1D fiber growth 50 µL of 2.51 nM (0.1 µg/mL) PCL 45 -b-PDMA 348 seed solution was spin-coated onto the cleaned coverslip twice at 3200 rpm for 50 seconds, immediately followed by 4000 rpm for 30 seconds. A cleaned plastic spacer was then placed onto the seed-coated coverslip to form a reaction chamber. This reaction chamber has then been mounted above the objective with the objective being adjusted to focus on the surface (Fig. ). PCL 73 -b-PDMA 204 unimer stock solutions were diluted in methanol to achieve the targeted concentration (0.06 µM, 1.67 µg/mL) and added into the reaction chamber, followed by iSCAT imaging immediately.
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| 26 |
To obtain data shown in Figs. & For other experiments, spin coating was used to prepare seeds-coated surface: 50 µL of moved from the chamber and the chamber was rinsed gently with methanol to remove unimer residues, followed by the addition of PCL 45 solution. This alternating sequence was repeated until the desired number of annuli was achieved. Recording started immediately after each unimer addition. A laser power density of 4 µW µm -2 at 637 nm, a camera exposure time of 400 µs, and an overall time-lapsed frame rate of 1 s -1 were selected. For the experiment in Fig. , PCL 45 :PCL 45 -b-PDMA 348 mixtures with sequentially increasing unimer concentration were added at each annulus. A laser power density of 4 µW µm -2 at 637 nm, a camera exposure time of 400 µs, and an overall time-lapsed frame rate 0.5 s -1 were selected.
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| 27 |
An inverted FV3000 (Olympus) confocal microscope with 20× and 60× oil immersion objectives was used for imaging. Scan rates of 1 µs pixel -1 at 512 by 512 pixels to 1024 by 1024 pixels were used. For platelet characterization, a minimum number of 100 platelets were segmented manually using Fiji image analysis software. iSCAT characterization A wide-field iSCAT was previously constructed (detailed in Fig. ). Briefly, a 637 nm multimode diode laser (RLM-6000L, Kvant lasers, Slovakia) was fiber-coupled before homogenization (Albedo system, Errol, France), generating a top-hat profile. Following collimation, the beam was focused at the back focal plane (BFP) of the objective (100× Plan-Apo 1.45 NA, Nikon, Japan). A polarizing beam splitter (PBS) transmits plane-polarized light with a specific orientation to the quarter-wave plate (QWP), which converts it to circularly polarized light. When used together, these components function as an optical isolator, efficiently separating the signal of interest from the illuminating light. Light reflected back from the sample-glass interface interferes with the light scattered by the objects within the sample, was then separated from the incident light, and directed onto a high-speed CMOS camera (PCO.dimax CS1, ExcelitasPCO GmbH, Germany). Focus control is then provided by a piezo-stage (P-545-3R8S, Physik Instrumente, Germany). The camera, laser, and piezo-stage are controlled using a custom LabVIEW program (National Instruments, USA).
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| 28 |
Glass coverslips (24×60 mm, #1.5 thickness, Epredia) were cleaned by sequential sonication in chloroform, acetone, and isopropanol for 15 minutes before drying with N 2 . Circular silicone spacers (ϕ9×2.5 mm thickness, Merck) were washed with the same protocol, dried under vacuum, and then placed on top of the cleaned coverslip to form a chambered coverslip (Fig. ). The reaction chamber was sealed with an additional coverslip (22×22 mm, Merck) which was cleaned using the same procedure. To prepare plasma-cleaned coverslips, the solvent-cleaned coverslips were treated with oxygen plasma for 6 minutes (Diener Electronic, Femto, Germany).
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| 29 |
PCL 45 :PCL 45 -b-PDMA 348 platelets synthesized in bulk were spin-coated onto the cleaned coverslip at 3200 rpm for 15 seconds until the desired surface density was achieved. Then a plastic spacer was placed onto the platelet-coated surface before the addition of methanol: water (30:70, v: v) solution and imaging. A laser power density of 2 µW µm -2 at 637 nm and a camera exposure time of 900 µs were selected unless otherwise stated. A minimum number of 100 platelets were analyzed to determine the average size.
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| 30 |
Image analysis of raw images collected from iSCAT microscopy experiments proceeded via ( were then added onto the seed-coated surface followed by recording immediately. A laser power density of 4 µW µm -2 at 637 nm, a camera exposure time of 400 µs, and an overall time-lapsed frame rate of 1.5 s -1 were chosen. This movie corresponds to the data in Fig. and.
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| 31 |
Background-corrected movie of the early growth stage of platelets with sizes below the diffraction limit (scale bar: 3 µm). 50 µL of 2.51 nM (0.1 µg/mL) seed solution was spin-coated onto a cleaned coverslip twice (3200 rpm for 50s followed by 4000 rpm for 30s). PCL 45 :PCL 45 -b-PDMA 348 mixtures in THF were then diluted with methanol to achieve a final concentration of 0.35 µM (3.33 µg/mL). 150 µL of unimer methanol solution was added onto the seed-coated surface and iSCAT observation started immediately. A laser power density of 4 µW µm -2 at 637 nm, a camera exposure time of 400 µs, and an overall time-lapsed frame rate of 10 s -1 were chosen. This movie corresponds to the data in Fig. and.
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| 32 |
Background-corrected movie of platelet growth recorded with high speed (scale bar: 2 µm). 50 µL of 2.51 nM (0.1 µg/mL) seed solution was spin-coated onto the cleaned coverslip twice (3200 rpm for 50 s followed by 4000 rpm for 30 s). PCL 45 :PCL 45 -b-PDMA 348 mixtures in THF were then diluted with methanol to achieve final concentration of 0.29 µM (2.78 µg/mL). 150 µL unimer methanol solution was added onto the seed-coated surface and iSCAT observation started from the mid-stage of platelet growth. A laser power density of 24 µW µm -2 at 637 nm, a camera exposure time of 80 µs, and an overall time-lapsed frame rate of 3000 s -1 were chosen. This movie corresponds to the data in Fig. and H.
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| 33 |
4. Stacks (Fig. ) were subtracted by a background corresponding to the medianaverage of around 10 frames corresponding to the image area prior to platelet growth (Fig. ), which generates the background corrected result as shown in Fig. . 7. After setting the measurement scale, particle parameters, such as area, long/short axis length, and aspect ratio were then collected using the built-in 'Analyze Particle' function in Fiji. iv
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Particles were selected based on the following filtering conditions: the diameter of the circle used to detect and analyze spots was set to 21 pixels; a minimum separation between particles of 2× this diameter was specified; the minimum integrated brightness was set to 0.01 and a threshold with the value of 0.001 was applied.
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Application of machine learning to chemical and physical problems is a growing field with applications ranging from discovery of new molecules and materials to improving existing theoretical models. The present article is about using machine learning to represent potential energy surfaces for small molecules and chemical reactions. The methods discussed in this article can be used for representing sets of coupled potential energy surfaces as required for simulating electronically nonadiabatic photochemical processes.
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Representation of coupled potential energy surfaces for multiple electronic states has been advanced by means of progress on diabatization methods, which have been reviewed recently. Diabatization enables the study of nonadiabatic processes with nuclear quantum dynamics or with large ensembles of long-time semiclassical nonadiabatic dynamics. Two methods are competing for adoption, namely neural networks and multilinear regression with high-order polynomials. From a supervised-learning perspective, although both are universal, neural networks are gaining a stronger foothold in the field due to their high convenience. However, a notorious drawback for machine-learned potentials is their unreliable behavior when applied to data quite different from that used for training. For example, it's easy for neural-network potentials to acquire unphysical holes or unphysical barriers and/or to have the wrong asymptotic behavior.
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Maintaining physical behavior in regions where one or more interatomic distance is very large or very small is hard because one does not usually saturate such regions of space with training data. When poor physical behavior arises, one often resorts to an iterative scheme: fit the surfaces, run trajectories to discover regions of space where the fit is unphysical, add data in the problematic regions, refit, and iterate the steps until no more unphysical behavior is found. However, this is problematic for maintaining physical behavior at large internuclear separation because the amount of space in asymptotic regions is infinite. And it can be troublesome in regions where two or more atoms get very close because the potential is very steep in those regions.
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In this work, we will introduce a new step in the machine-learning procedure, namely, a parametrically managed activation function. This is designed so that the potential representation in difficult regions inherits the correct physical behavior of lower-dimensional potentials. We will introduce the parametrically managed activation function in the context of the methods of diabatization by deep neural network (DDNN) and permutationally restrained DDNN (PR-DDNN), but the type of approach employed here should also be useful for other uses of neural networks to represent potential energy surfaces. Section 2 reviews the relevant aspects of the DDNN and PR-DNN schemes. Section 3 presents the new method. Section 4 illustrates the new method by using it to obtain improved potential energy surfaces for O + O2 collisions in the 3 A´ electronic state manifold.
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We first introduce diabatization by deep neural network (DDNN). The DDNN method is based on a diabatic potential energy matrix (DPEM), whose eigenvalues are the adiabatic potential energies. The diagonal elements 𝐔 !,## of the DPEM at geometry j are the diabatic potential energies, and the off-diagonal elements 𝐔 !,#$ are called diabatic couplings. Because the elements of the DPEM are smooth and the adiabatic potentials are not smooth, the strategy adopted is to fit the DPEM and recover the adiabatic potentials by diagonalization. An For an L-layer DDNN, the input layer for geometry n is 𝐚 ) * , the hidden layers are 𝐚 ) + to 𝐚 ) ,-+ , the DPEM layer is 𝐚 ) ,-* , and the adiabatic potential energy layer is 𝐚 ) , . The propagation from the input layer to hidden layers is done by
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6449ae54df78ec501570cef6
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) where a superscript T denotes a transpose, 𝐰 . and 𝐛 . are respectively the connection weights and biases of layer l, and 𝑔 is the hidden-layer activation function. Nonlinear hidden layer activation functions like rectified linear unit (ReLU) or Gaussian Error Linear Unit (GELU) are suggested to prevent the vanishing gradient problem. Note that GELU is an improved version of ReLU with a global continuous behavior. The GELU activation function is defined as
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| 6 |
Optimization of the weights and biases according to eq (4) does not ensure that the adiabatic energies are invariant to permutation of identical nuclei. To promote permutational invariance, we add an additional database called permuted adiabatic database 𝒮. Database 𝒮 has 𝑆 = 𝑁 8 𝑁 geometries, where 𝑁 8 is the number of non-identity permutations of identical nuclei. For example, in O3, there are three transpositions and two cyclic permutations of O atoms, and hence 𝑁 8 = 5. For each geometry s in the permuted adiabatic database 𝒮, the target adiabatic potential energies 𝐕 9 are known because they are identical to those for the unpermuted geometries in database 𝒩.
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Note that an alternative to the above would be to constrain the permutation invariance of identical nuclei rather than to restrain it. This could be accomplished by using permutationally invariant input for input layer 𝐚 ) * (which constrains permutation invariance in the input stage), for example permutationally invariant polynomials or fundamental invariants, or by using an equivariant neural network that constrains permutation invariance in the output stage. Both approaches constrain the permutational invariance of adiabatic potential energies as well as DPEM elements. However, DPEM elements are not necessarily permutationally invariant, and the PR-DDNN method allows the combination of permutational variance of the diabatic representation with permutational invariance of the adiabatic one. Furthermore, the PR-DDNN method recognizes that one does not fit the unpermuted adiabatic data perfectly and allows imperfect fitting of permutational invariance, which is a balanced approach, rather than insisting that exact permutational invariance be combined with inexact fitting of adiabatic potentials.
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The global potential energy surface of a many-body system often reduces in dimensionality at the dissociation limit. For example, when O is separated from O2, an O3 global adiabatic potential energy surface, which depends on three internal coordinates, should reduce to a onedimensional diatomic potential energy surface of O2 (that depends on the electronic state of O2) plus a constant (which is the energy of a given state of O). To treat dissociation, we assume the entire system Z is composed of subsystems X and Y, and we considering the reaction
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We let 𝐑 = denote the entire set of coordinates, we let 𝐑 > denote the subset of coordinates pertaining to X, and we let 𝐑 ? denote the subset of coordinates of Y. The adiabatic potential energy 𝐸 = (𝐑 = ) of Z at geometry RZ can be written for a single electronic state as
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6449ae54df78ec501570cef6
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where 𝐸 > (𝐑 > ) is the adiabatic potential energy of X at geometry 𝐑 > , 𝐸 ? (𝐑 ? ) is the adiabatic potential energy of Y at geometry 𝐑 ? , and 𝐸 @2 (𝐑 = ) is the interaction potential energy, which will be called the added-dimensions (AD) potential energy because that is a more suitable name for the general case introduced in Section 3.2. Therefore, at the dissociation limit where X and Y are separated, the added-dimensions potential energy 𝐸 @2 (𝐑 = ) is equal to 0. Let nX and nY be the number of atoms in X and Y. Since the potential energy depends only on internal coordinates, 𝐸 > (𝐑 > ) + 𝐸 ? (𝐑 ? ) has a lower dimensionality than 𝐸 = (𝐑 = ), and we label this sum as the lowdimensional (LD) adiabatic potential energy 𝐸 A2 (𝐑 = ). For example, if X and Y are both polyatomic, then
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| 11 |
Note that 𝐄 = (𝐑 = ) is not the final adiabatic potential energy vector of Z because the elements are not sorted in energetic order. In practice, our preferred choice of diabatic representation usually satisfies the constraint that at any dissociation geometry (where 𝐑 = equals 𝐑 =,G ), the diabatic potentials equal the adiabatic ones, although they might be in a different order. We will assume this usual constraint in the present article. However, we will also require the following constraints that include state ordering:
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Next, we recast the formalism to treat it any low-dimensional potential that vanishes or may be assumed to vanish when the subsystems separate, i.e., the LD potential need not be the sum of the potential of X and the potential of Y, and the AD potential need not be the interaction between two subsystems. We let 𝐑 A2 denote any geometry where the system is dissociated or where the AD potential is to be approximated as zero. We employ a diabatic representation in which
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where 𝚵 @2,! is a vector of unique matrix elements of 𝐔 @2 defined in eq (19). Therefore, we are effectively using the feedforward neural networks of DDNN and PR-DDNN to fit the interaction DPEM 𝐔 @2 . We add the low-dimensional DPEM 𝐔 A2 at the DPEM layer, which is layer L-1.
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Diagonalizing the DPEM yields the final adiabatic potential energy. Constraining the potential of the high-dimensional system to be a function of the low-dimensional subsystems in the dissociation limit means that DDNN and PR-DDNN predict zero vectors for DPEM layer for dissociation limit geometries, 𝐚 )∈LD ,-* = 𝟎 (24) where "𝑛 ∈ LD" denotes that geometry n corresponds to the dissociation limit.
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One possible application of the more general formalism presented in this section is to treat dissociation when the LD potential is not simply the sum of two fragment potentials and hence the AD potential is not simply the interaction potential. An example of this kind of application is discussed in Sections 3.3 and 4. In that example, we apply the theory where LD is the two-body approximation, and AD is the many-body correction to the two-body approximation.
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Another example of an application where this kind of treatment can be useful is the region where two atoms get close. In such regions, the potential is very steep, and any analytic fit extrapolated into this region is liable to have wells instead of steep repulsive wall. However, a system with nZ atoms has nZ(nZ -1)/2 internuclear distances. Because this is quadratic in the number of atoms, it is often prohibitive to include data in all regions where two atoms get close.
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However, one often does not need quantitative accuracy in regions where the repulsive interaction is very large; one simply needs to be sure that the analytic potential is repulsive enough to keep trajectories from unphysically entering such a region. A pairwise additive potential is repulsive in any such region, and designing the full potential be a pairwise potential plus corrections is a reasonable strategy. Then, having the full potential reduce to a pairwise additive repulsive potential in highly repulsive regions without having data there would be a convenient way to ensure a physical fit at close distances. Note that for nZ > 4, nZ(nZ -1)/2 is greater than 3nZ -6, so a pairwise additive potential does not depend on less coordinates than 3nZ -6, but rather it is a sum of lower-dimensional potentials (in this case a sum of 1dimensional potentials), and that is what we signify when we label it as LD.
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As mentioned in the introduction and in Section 3.2, we anticipate two kinds of region where the potential can be set to the LD potential. One is the region where two subsystems dissociate and another region is where two atoms get very close. To illustrate the use of a parametrically managed activation function, we next give an example suitable for treating the dissociation case. Section 3.1 converted the problem of achieving a physical extrapolation to asymptotic regions into the problem of how to satisfy eq (24). In regression problems, the output layer often employs the identity function as an output activation function as shown in eq (3). To satisfy eq (24), we generalize eq (3) to 𝒛 ,-* = (𝐰 ,-+ ) 0 𝐚 ) ,-+ + 𝐛 ,-+ (25)
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Now we combine PR-DDNN with separation of the interaction potential from the lowdimensional potential with the SBW parametrically managed activation function for semiautomatic simultaneous diabatization and adiabatic surface fitting. We will call the neural network that combines these features a parametrically managed DDNN or PM-DDNN. The PM-DDNN forces the global potential energy surfaces to behave as low-dimensional potentials at dissociation-limit geometries.
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We set a equal to a negative number, in particular a = -2.0 a0, because we are only concerned with large values of 𝑟 IJ%%,) in the present application. (Note: 1 a0 ≡ 1 bohr ≈ 0.5292 Å.) We set f equal to 2.0 a0 -1 and b equal to 7.5 a0 because these choices of the parameters gave physical fits.
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In the current work, 𝛼 * = 1.0, and 𝛼 + = 0.0001. The input adiabatic database contains the same 6340 geometries as described previously. We tested several node structures of PM-DDNN corresponding to different numbers of hidden layers and hidden layer neurons (HLNs). We made several runs with each combination, and the best mean unsigned errors (MUEs) of the unpermuted energies obtained for each node structure of PM-DDNN are provided in Table . The first and last number in the structure column are fixed because the first number corresponds to the number of input coordinates, and the final number corresponds to the number of unique DPEM elements. For the present 14-state system, we have 105 unique elements of the DPEM. The final selection of global surfaces involves a balanced compromise between accuracy and efficiency because a larger number of HLNs usually improves the accuracy but always raises the cost of evaluating the potential energies, couplings, and gradients during simulations.
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The final surface has a root-mean-square error of the adiabatic database (the unpermuted geometries used in the first term of the cost function) of 106 meV and a root-mean square error of the permuted adiabatic database (the last term of the cost function) of 126 meV. The closeness of these numbers shows that the restraint on energies at permuted geometries was successful.
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The mean unsigned error (MUE) of the unpermuted adiabatic database is 61 meV, and the mean signed error on the permuted adiabatic database is 68 meV. The smallness (106/61 = 1.74 and 126/68 = 1.85) of the ratios of the root-mean-squared errors to the MUEs is an indication that there are not an excessive number of large outliers.
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| 24 |
In the long run, when the goal is a global potential energy surface, it is irrelevant whether a geometry is found in the permuted or unpermuted set, and we simply want a reasonably small error on all adiabatic energies. Table provides further details of the MUEs of the entire set of 532560 adiabatic energies, breaking them down by electronic state and energy range. For each state in each energy range, the table shows the number of geometries and the MUE. At energies below 10 eV, the MUE is not a strong function of state. Close examination of the table shows remarkably good across-the-board accuracy up to 10 eV and quite reasonable relative accuracy at higher energies.
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The desirable new property of PM-DDNN with the parametrically managed activation function of Section 3.3 is that it damps the interaction potential to zero at large separation distances. This builds physical extrapolation into the results of the neural network. To assess this property for the results obtained by PM-DDNN, we computed GTrV𝐔 @2 (𝐑)WG, and and d indicate the regions with JTrV𝐔 @2 (𝐑)W Trk𝐔 A2 (𝐑)m y J < 0.1%. We conclude from Figure that PM-DDNN achieves a physical fit at large separation distances, which was the primary objective of the present work.
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In the future, one can explore other strategies. For example, in a triatomic system where one is concerned with managing potential representations in asymptotic regions, one might base the parametrically managed activation function on the second largest interatomic distance (rather than the largest). For another example, in any polyatomic system where one is concerned with managing potential representations in highly repulsive regions, one might base a parametrically managed activation function on any or all interatomic distances that become very small.
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We illustrated the method for the former case by presenting a parametrically managed activation function that is specifically designed to damp interaction potentials toward zero in asymptotic regions -even without placing data there. The parametrically managed activation function depends parametrically on the separation distance of the fragments of the full system. The result is an improved set of 14-state 3 A´ global potential energy surfaces for O + O2 collisions that
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shows that the neural network potential can be used safely outside the range of its data. Although the equations in present article were written in the language of a low-dimensional potential, one could also apply this method where the low-dimensional potential is replaced by a lower-level potential with the same dimensionality as the final potential.
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We emphasize that the step of using a parametrically managed activation function can also be employed in other neural network architectures and contexts. One of the disadvantages of neural network potentials, as compared to ones designed with more human input, has been that they do not necessarily augment the mathematical steps with underlying human understanding, for example, by choosing a functional form that satisfies the known characters of the potentials.
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The same 𝑉 2C(RS) (𝑟 M ) is used for all fourteen states, and it is the same in the 2023 potentials as in the 2022 ones. For 𝑉 1 P; (𝑟 M ), the 2023 potentials use the same functional form as was used in the 2022 potentials, but we improved the parameters by calculating extra points in the vicinity of the potential minima and adding these points to the diatomic data used for fitting, The SR functional form is an even-tempered Gaussian function:
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60c73eda469df4affff42967
| 0 |
Proteochemometrics or quantitative multi-structure-property-relationship modeling (QM-SPR) is an extension from the traditional quantitative structure-activity relationship (QSAR) modeling. In QSAR, the target protein is fixed and its interaction with ligands (small molecules or compounds) is predicted only from ligands descriptors. On the contrary, the aim of proteochemometrics is to predict the binding a nity value by modeling the interaction of both proteins and ligands. For this, a data matrix is built, each of its rows containing descriptors of both target and ligand linked to some experimentally measured biological activity. A statistical or machine learning method is then used to induce the model. The main advantages over QSAR are twofold: first, that the induced model can be applied for predictions of interaction with new proteins as well as ligands and second, that it can consider the underlying biological information carried by the protein as well as other possible cross-interactions of the ligand.
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60c73eda469df4affff42967
| 1 |
Deep learning (DL) is a branch of machine learning that stems from artificial neural networks, which are computational models inspired in the structure of the brain and the interconnection between the neurons. DL is able to learn representations of raw data with multiple levels of abstraction. These concepts started to be developed in the 1940s 3 but it was not until 2012 that there was a break through of the Deep Neural Networks (DNN). Since then, DL has been successfully applied in natural language processing, 5 image recognition, drug discovery 7 or computational biology. The increase of computational power by parallel computing with graphics processing units (GPU) and the improvement of optimizers and regularization techniques contributed to this resurgence, along with the development of software platforms that allow to make prototyping faster and automatically manage GPU computing, like Theano or Tensorflow. DL provides a framework for the identification of both of biological targets and biologically active compounds with desired pharmacological e↵ects. In 2012, DNN won a QSAR machine learning challenge on drug discovery and activity prediction launched by Merck, outperforming Merck's Random Forests baseline model by 14% accuracy. Since then, the application of DL to pharmaceutical problems gained popularity, although it has been mainly applied to multitask QSAR modeling. Regarding DL-based proteochemometrics, little has been done except for the work of Lenselink et al, where they compared di↵erent machine learning methods for proteochemometrics, being DNN the top performer.
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The predictive power of a consistent model must remain stable when applied to data that comes from a di↵erent source than the training set. Moreover, possible redundancy in the data must be controlled. Proteins are divided into families, which usually have similarities in sequence or structure. Compounds might be part of the same chemical series. The performance of classification model should be tested when applied to families of proteins or compounds with di↵erent sca↵olds than those used to train it. On the latter, Wallach and Heifets concluded that performance of most of the reported ligand-based classification problems reflect overfitting to training benchmarks rather than good prospective accuracy, mainly because of the redundancy between training and validation sets. This issue becomes more critical when using random cross-validation: in the pharmaceutical field, compounds are usually synthesized serially to enhance molecular properties. This leads to training and validation sets following the same distribution, which is desirable in most machine learning problems, but a poor estimate of reality in drug discovery. Time-split validation is common practice in pharmaceutical environment to overcome this issue. This strategy is well suited to the realistic scenario, where we are interested in prospective performance of the models. However, most public data lack of temporal information, hindering this strategy to be applied. Additionally, time-split data has also shown to be biased because of the high similarity between discovered actives in di↵erent phases. Other techniques have been applied to reduce bias and data redundancy between training and validation sets. Unterthiner et al. clustered compounds using single linkage to avoid having compounds sharing sca↵olds across training and validation sets. Rohrer and Baumann designed the Maximally Unbiased Validation (MUV) benchmark to be challenging for standard virtual screening: actives have been selected to avoid biases of enrichment assessment and inactives have been biologically tested against their target. 32 Xia et al. presented a method to ensure chemical diversity of ligands while keeping the physicochemical similarity between ligands and decoys. Wallach et al. removed analogue bias in active molecules by clustering and selected decoys to match in sets to actives with respect to some 1D physicochemical descriptors while being topologically dissimilar based on 2D fingerprints. However, these unbiasing techniques only focus on redundancy between actives, overlooking the impact of inactive-active or inactive-inactive similarity, which leads to models memorizing the similarity between benchmark inactives and hence, overfitting. Another related issue is that the possible bias across the di↵erent data sources used in some studies has not been properly studied yet. Di↵erent datasets might have di↵erent structure, a↵ecting to the study of the generalization of the model. A related issue is found in the study of Altae-Tran et al, where after the collapse of their transfer learning experiments it is a rmed that one-shot learning methods may struggle to generalize to novel molecular sca↵olds, and that there is a limit to their cross-tasks generalization capability.
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Analysis of bias in binding classification models have been always focused on QSAR models, but how could a↵ect the inclusion of proteins to bias in QMSPR models remains unknown. Proteins are macromolecules constituted by amino acid residues covalently attached to one another, forming long linear sequences which identify them, defining its folding and its activity. The main value of DL in this context is that DL can directly learn from the sequence, capturing nonlinear dependencies and interaction e↵ects, and hence providing additional understanding about the structure of the biological data. The appropriate DL architecture to manage this kind of data are bi-directional Recurrent Neural Networks (RNN), well suited for modeling data with a sequential but non-causal structure, variable length and long-range dependencies. Baldi et al have applied bi-directional RNN to protein sequence for predicting secondary structure, for matching protein beta-sheet partners or for predicting residue-residue contact. However, classical RNN cannot hold very long-range dependencies and to overcome this issue Hochreiter et al applied Long Short Term Memory (LSTM) networks to classify amino acid sequences into superfamilies. Jurtz et al applied bi-directional LSTM to amino acid sequence for subcellular localization, secondary structure prediction and peptides binding to a major histocompatibility complex. In this paper, we analyse and quantify the e↵ect of di↵erent cross-validation strategies on the performance of binding prediction DL-based proteochemometrics models. Additionally, we compare these DL models with baseline logistic regression (LR) models and explore di↵erent representations for molecules.
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Models were trained on the dataset generated from three di↵erent publicly available sources by Riniker and Landrum for true reproducibility and comparability of benchmarking studies. This dataset incorporates 88 targets from ChEMBL, the Directory of Useful Decoys (DUD) and the MUV. The selection of actives and decoys was conducted on drug-like molecules and in such a way as to cover the maximum range of the chemical spectrum, based on diversity and physical properties. ChEMBL and DUD decoys were selected from the ZINC database. The selection of ChEMBL targets was based on the 50 human targets and actives proposed by Heikamp and Bajorath study and performed on ChEMBL version 14. We only selected 500 decoys randomly from all those available for each target, in order to have a more computationally-approachable dataset and to decrease active/decoy imbalance per target while keeping a plausible proportion. The list of molecules identified by their SMILES was then standardized to avoid multiple tautomeric forms. Finally, these compounds were filtered to remove salts, those with molecular weight >900Da or >32 rotatable bonds and those containing elements other than C, H, O, N, S, P or halides.
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We tested two ways of representing input molecules: (1) as sequences of symbols, using the SMILES notation and (2) as the combination of molecular fingerprints, where structural information is represented by bits in a bit string. The SMILES representation as input for a DL model was based on the DeepCCI by Kwon et al. Model input has to be numerical, so SMILES notation was one-hot encoded (Figure ). This means that every character of the SMILES string was represented by a binary vector of size 35, with all but its corresponding entry set to zero. SMILES were padded to the length of the longest string, 94.
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For the fingerprints representation we selected three of them: topological torsions (TT) fingerprint, extended connectivity fingeprint and functional connectivity fingerprint, both with a diameter of 6 (ECFP6 and FCFP6, respectively) 51 (Figure ). TT describe four atoms forming a torsion, and the atom type includes the element, the number of nonhydrogen neighbors and number of ⇡ electrons. ECFP6 and FCFP6 encode circular atom environment up to 6 bond length. In ECFP6, atom type includes the element, the number of heavy-atom neighbors, the number of hydrogens, the isotope and ring information. FCFP6 use pharmacophoric features. All of them were generated using the RDKit package, and defined with a length of 1024 bits, since there is proof of a very low number of collisions with this size. For protein representation, raw amino acid sequences were fed to the model (Figure ). As for SMILES strings, these sequences were converted to numerical through one-hot encoding, only that in this case each amino acid was represented by a binary vector of length 20. Amino acid sequences were then padded to the length of the longest target, in this case, 1988.
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Four di↵erent cross-validation strategies were applied to both active and inactive Riniker dataset compounds (see Figure ), omitting binding targets. In all cases active/inactive A. validation/test set and thus control for the compound series bias. Clusters were randomly joined and assigned to the splitting sets in order to have 80/10/10 splitting (Figure ). ( )
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Intermediate, where the previous K-Means clustering was also applied, but only to those compounds coming from ChEMBL (Figure ). In order for this schema to have a test set of comparable size with the others, only one data source was used. We chose MUV dataset since it was designed to be challenging, as seen in the Introduction, while data architectural design of the original DUD is not that well suited for this problem. In Figures , S4 and S5 of the Supporting Information, the proportion of actives/inactives for each target in each splitting set is depicted for random, clustering, database-based and intermediate cross-validation strategies, respectively.
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This baseline model consisted on two input layers concatenated, one for the compound and one for the target, with as many neurons as the size of the input (94 and 3072 in the case of SMILES and fingerprints, respectively and 1988 in the case of targets) connected to a sigmoidal unit (see Figure ).
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An schematic representation of the DL predictive models used can be seen in Figure and. The amino acid sequence analysis block is common for both models and it is a Convolutional Recurrent Neural Network based on the one used by Jurtz et al for the prediction of subcellular localization of proteins 42 (Figure ). This architecture allows to build complex representations from both targets and compounds for the prediction of binding, in contrast to the LR models, which are directly fed with the input features. The input of the model is the amino acid sequence one-hot encoded, which is passed through a 1D Convolutional Neural Network (CNN). This 1D CNN comprises filters of sizes 1, 3, 5, 9, 15 and 21, with the aim of detecting motifs of di↵erent length in the amino acid sequence. Convolutional layers are followed then by a max pooling layer, downsampling the input and thus, reducing the number of model parameters. The input is then introduced to a bi-LSTM neural network. Dropout algorithm is used in di↵erent parts of the model to prevent overfitting. The compound analysis block depends on the encoding of molecules. If molecules are represented by their SMILES, then the compound processing block is similar to the sequence processing block (Figure ). The SMILES string is one-hot encoded and the input is passed to a bank of convolutional filters, in this case of size 3,4 and 5 based on the sizes of the LINGO substrings analysed by Vidal et al. After that, a maximum pooling layer condenses information and transfer it to a LSTM, but in this case uni-drectional since the SMILES strings are causal, in the sense that they are read in only one direction. Dropout is also used here to prevent overfitting. If molecules are represented by ECFP6, FCFP6 and HASHTT fingerprints, namely a binary vector of length 3072, the input is passed through a feed-forward neural network followed by dropout (Figure ).
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Finally, the sequence and the compound analysis blocks are merged and the information is processed by a sigmoid activation unit, which quantifies how likely the sequence-compound binding is. Binary predictions are obtained thresholding the activation at 0.5. All the models (both DL and LR) were trained with Adam optimizer 10 for 500 epochs, with a batch size of 128 for training and 64 for validation (learning rate=5e-6 for DL models encoded by fingerprints, 5e-5 for the rest). Decay rate was defined as learning rate/number of epochs.
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Each one of the cross-validation strategies is analyzed in terms of imbalance and data redundancy to better understand and interpret performance results. First, active/inactive proportion is explored for each cross-validation schema. Then, overlap of targets and compounds between split sets is computed as a percentage with respect to the total number of targets (88) and compounds (32,950), respectively. Lastly, distribution of chemotypes and protein classes is explored for each strategy. For targets, this distribution is studied for the main protein families. Since for molecules there is no such classification, we decided to group them in terms of their Bemis-Murcko sca↵old (BMS), a technique for extracting molecular frameworks by removing side chain atoms which has been used for clustering compounds. Performance metrics
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Area under the receiver operating characteristic (ROC) curve (from now, referred to as AUC), traditionally employed for measuring the performance of classification models, has been reported for not being enough for evaluating virtual screening models because it is not sensitive to early recognition and it is a↵ected by class imbalance. Thus, we complement this information with partial AUC (pAUC) at 5%, which allows to focus on the region of the ROC curve more relevant for virtual screening 1 (up to 5% of the False Discovery Rate), with Cohen's kappa coe cient (), which measures the agreement between real and predicted classification, and with the Boltzmann-enhanced discrimination of receiver operating characteristic (BEDROC), a metric proposed to overcome the limitations of AUC 57 increasingly popular in the evaluation of virtual screening models. BEDROC uses an exponential function based on parameter ↵ and is bounded between 0 and 1, making it suitable for early recognition. As recommended by Riniker et al, we focus on AUC and BEDROC with ↵=20, whilst also reporting ↵=100.
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We implemented and trained DL and LR binding classification models. We then selected the best training epoch in terms of F1 Score, the harmonic mean of precision and recall, on the validation set, since it can handle class imbalance. Finally, we tested the selected models on the corresponding test set of each cross-validation strategy. Stratified subsampling of the 80% of the test data was used to sample 100 values from the performance estimates distributions. The nonparametric Wilcoxon rank-sum test was used to compare AUC and BEDROC (20) metrics between all pairs of models. P-values were adjusted for multiple testing by computing the False Discovery Rate (FDR) by Benjamini-Hochberg 60 for each metric.
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In clustering-based and random, every target is repeated on the three splitting sets. In intermediate we only find overlapping targets between training and validation, since both splitting sets are built from the ChEMBL dataset, while test corresponds to the MUV database. In database-based, there is no overlap of targets between splitting sets.
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Figure shows overlap of compounds between splitting sets for each cross-validation schema. In this case, in clustering-based there are no repeated compounds across groups. In database-based and intermediate there is negligible overlap between groups, probably due to repeated inactives for di↵erent targets. Since in random cross-validation splitting was made randomly, there are repeated compounds in all splitting sets.
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A summary of performance metrics of each model can be seen on Table . are not conclusive on which one performs better. In terms of AUC, the best performance is from a logistic regression model and in terms of BEDROC, a deep learning model. In general, the best algorithm depends of the cross-validation strategy and the compound representation, but there is a tendency of logistic regression being better for the SMILES representation of the compound, and deep learning for the fingerprints representation.
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In Figure , performance of the di↵erent cross-validation strategies is better depicted: the values of AUC and BEDROC (20) for each schema, compound representation and algorithm are compared between them and with a random prediction. Error bars show the standard deviations obtained from subsampled estimates. It can be seen that random strategy outperforms the rest in all the possibilities. After random, clustering-based strategy had the best performance both in terms of AUC and BEDROC (20). Database-based and intermediate strategies have both poor performance, specially in terms of BEDROC (20). The same behavior can be seen in ROC curves of all possibilities in Figure of the Supporting (20) metrics between pairs of models. Di↵erences are calculated subtracting column performances from rows. Crossed tiles indicate that difference for that pair of models is not statistically significant (↵ = 0.05).
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The mean di↵erence of BEDROC(20) metrics for each possible combination of models is shown in 10. Di↵erences in terms of AUROC follow the same behavior but of a smaller magnitude (see Figure of Supplementary Information). The most remarkable di↵erences can be seen on random models versus database-based and intermediate models. There are also relevant di↵erences on performance between clustering-based cross-validation and the rest of strategies. families, while controlling for the data redundancy issue as the number of sca↵olds shared between train and test decreases from 2,676 (random) to 275. It also leads to less optimistic performance estimates than the random strategy, whereas the models still retain predictive power. Therefore, the clustering strategy is chosen as our reference, in line with previous studies.
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In general, models for compounds represented by their fingerprints outperformed models for SMILES representation in clustering-based and random strategies, for both AUC and BEDROC (20), see Figure . This can be due to the specific architectures employed in each case: the compound analysis block for SMILES-encoding is based on CNN and LSTM to capture the sequence structure, while the compound analysis block for fingerprints is based on a single feed-forward neural network. This di↵erence in model complexity and by extension, in the number of parameters, could have resulted in a poorly fitted SMILES-encoded model.
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Regarding the algorithm used within the clustering strategy and FP encoding, DL and LR appear technically tied at a BEDROC(20) around 0.53 (Table ). LR outperforms DL in terms of AUC (0.84 versus 0.74), but falls behind in the early recognition metrics pAUC(5%) (0.015 versus 0.017), BEDROC(100) (0.55 versus 0.72) and other metrics such as Cohen's Kappa (0.360 versus 0.445) and F1 score (0.36 versus 0.68). In agreement with previous studies, the AUC appears misleading for early recognition. Despite the tie between LR and DL, the alternative metrics favor DL and is therefore preferred over LR by a small margin.
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On the other hand, performance of the random cross-validation fingerprints-based DL model (BEDROC(20) of 0.89) is slightly lower to the other published DL-based proteochemometrics model 28 (0.96). Lenselink et al represent proteins through standard physicochemical descriptors, whereas we use their amino acid sequence. The fact that amino acid-based representations attain a good predictive power poses the opportunity to gain insights into the mechanisms causing protein-ligand binding by analyzing biological patterns in the CNN filters and the long-range dependencies in the LSTM. Regarding the validation, Lenselink et al apply a temporal split strategy that drops the BEDROC metric 0.11 units, while our clustering strategy penalizes 0.33 units to our DL model. This is expected as time-split cross-validation can still su↵er from chemical series bias.
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We have benchmarked protein-compound binding models using two molecular representations for compounds and two prediction algorithms under four cross-validation strategies. One of our main findings is the existence of a database-specific bias that challenges the generalization of machine learning models between databases. Performance estimates derived from classical random cross-validation are overly optimistic, despite being widely used in literature. Instead, we recommend a clustering-based cross-validation since it addresses the chemical series bias while providing more reliable performance estimates. For molecular representation, fingerprints have led to better models than the SMILES identification string.
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Thanks to improvements in algorithms, force fields, and computer hardware, Molecular Dynamics (MD) simulations have become a versatile tool for investigating the conformational landscape of complex biomolecular systems at the atomic level. An important algorithmic improvement has been the explicit inclusion of pH in MD simulations, as pH is an important experimental parameter that affects the structure and dynamics of biomolecules.
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To provide the users of the GROMACS MD package 13 with access to simulations at constant pH, we have implemented the λ-dynamics based constant pH approach by Brooks and co-workers. In contrast to the previous implementation in a fork of GROMACS 3.3, 10 the new implementation, which is described in an accompanying paper, 15 is efficient and constant pH MD simulations can be performed with little computational overhead compared to normal MD simulations. The purpose of this paper is to provide users with guidelines and recommendations on how to set up and perform constant pH MD simulations, including the necessary parameterization steps.
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In constant pH MD simulations, titratable groups can dynamically change their protonation state. These changes are driven by interactions between the group and the chemical environment, modeled with a force field, and the aqueous proton concentration, modeled with a pH potential. Because at the force field level, a number of contributions to the free energy of (de)protonation are not included explicitly, i.e. quantum mechanical interactions associated with bond breakage and formation, as well as the actual proton particle, corrections to the force field are needed in λ-dynamics based constant pH MD. In GROMACS these corrections are implemented as analytical functions, V MM (λ j ), fitted to the free energy profile associated with the deprotonation of a titratable residue j at the force field level. The accuracy of such free energy profiles not only depends on how closely the force field model represents the true potential energy surface but also on the convergence of sampling of all other degrees of freedom in the system. Therefore, whereas in normal MD, the accuracy of the dynamics depends solely on the quality of the force field, the accuracy of λ-dynamics based constant pH MD depends additionally on whether all relevant degrees of freedom are sampled sufficiently in the simulations required for parameterizing the correction potentials.
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We found that insufficient sampling of the dihedral degrees of freedom in the amino acid side chains can lead to poor convergence in the deprotonation free energy profiles, as was also observed by Klimovich and Mobley in simulations without constant pH. We traced the lack of the dihedral sampling to the barriers that separate the minima in the torsion potentials. These barriers are too high to reach a converged sampling of the dihedral free energy landscape on the timescales of typical constant pH MD simulations. Because the interaction between the titratable group and the environment depends critically on the dihedral angles of the side chain, a lack of convergence in these dihedral angles also affects the sampling of the protonation states.
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Rather than increasing the timescale of the MD simulations to obtain converged dihedral and protonation state distributions, or introducing enhanced sampling techniques, we propose to reduce the barriers for dihedral rotations in a systematic way. We will demonstrate that such optimized dihedral force field parameters improve pK a estimates of amino acids, without compromising the overall conformational sampling of the protein.
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With a higher accuracy of the underlying deprotonation free energy profiles, we found that the correct sampling of protonation states also depends critically on the order of the polynomial fit used to obtain an analytical form for these correction potentials. We show that the commonly accepted first-order fit, 12 although firmly based on linear response theory, is not sufficiently accurate and can lead to erroneous protonation dynamics in constant pH MD simulations.
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Constant pH MD simulations have been performed with various electrostatic models, including generalized Born, 9 shifted cut-off, and Particle Mesh Ewald (PME). Of these methods, the Ewald summation based PME method is generally considered to provide the most accurate description of the electrostatic interactions in periodic biomolecular systems. Because Ewald summation can only provide accurate results if the simulation box remains neutral, the charge fluctuations associated with the dynamic protonation and deprotonation in constant pH MD simulations need to be compensated to prevent artifacts. Titratable sites can be directly coupled to special particles, modeled as ions or water molecules, such that charge is transferred directly between the titratable site and that particle. Alternatively, all sites can be coupled collectively to a sufficiently large number of buffer particles. The latter approach has the advantage that spontaneous fluctuations in the interaction of the buffer particles with their environment affect all titratable sites to the same extent. The disadvantage is that the setup and parameterization of the buffer approach are more involved, as it requires selecting the number of buffers and parameterizing their interaction with the rest of the system. To facilitate the use of buffers in constant pH MD, we provide a parameterization strategy aimed at preventing buffer clustering, buffer binding to titratable sites, and buffer permeation into hydrophobic regions. We demonstrate that buffers parameterized with this strategy also avoid finite-size effects associated with the periodicity of small simulation boxes.
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The original and modified (described in detail below) CHARMM36m force fields were used in all simulations. The table also presents the box size, the number of CHARMM36 TIP3P water molecules, ions, and buffer particles included in each system. The fitting coefficients of the V MM correction potential for the buffer particles were obtained with system BUF 1 . To find the optimal charge range and Lennard-Jones parameters for the buffer particles, the enhanced sampling simulations with the accelerated weight histogram (AWH) method were performed on system BUF 2 . Systems ADA, AEA, AKA, and AHA are alanine tripeptides with capped termini, and as central residue aspartic, glutamic, lysine, and histidine amino acids, respectively. AAA 1 and AAA 2 systems are alanine tripeptides with protonated termini. C-and N-termini were made titratable in AAA 1 and AAA 2 systems, respectively. Two sets of simulations of the cardiotoxin V protein were performed. System 1CVO 1 was used to calculate the pK a values of titratable residues, while the larger system 1CVO 2 was used to compute the radial distribution function of the buffer particles around the protein. The membrane systems MEMB 1 and MEMB 2 contained 106 1-palmitoyl-2oleoyl-glycero-3-phosphocholine (POPC) lipids. Starting coordinates and topologies of all systems are provided as Supporting Information.
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Periodic boundary conditions were applied in all systems. Electrostatic interactions were modeled with the particle mesh Ewald method, while van der Waals interactions were modeled with Lennard-Jones potentials which were smoothly switched to zero in the range The leap-frog integrator with an integration step of 2 fs was used. Bond lengths to hydrogens in the solute were constrained with the LINCS algorithm, while the internal degrees of the CHARMM TIP3P water molecules were constrained with the SETTLE algorithm. Prior to the constant pH MD simulations, the energy of all systems was minimized using the steepest descent method, followed by a 1 ns equilibration.
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In the constant pH MD simulations, the mass of λ-particles was set to 5 atomic units, and temperature was kept constant at 300 K with a separate v-rescale thermostat for the λ-degrees of freedom, with a time constant of 2.0 ps -1 . The single-site representation, defined and described in the accompanying paper, 15 was used for Asp, Glu, Lys, C-ter, and N-ter, whereas the multisite representation, also described in that paper, was used for His. The same pH and biasing potentials were used as in Aho et al. In the sampling simulations of single titratable residues, the pH was set equal to the pK a , and the barrier height of the biasing potential was set to zero. The titration of the cardiotoxin V (PDB ID: 1CVO ) protein was performed by running ten independent replicas of 100 ns each, for 15 equidistantly spaced pH values in the range from 1.0 to 8.0, using both the original and modified CHARMM36m force field. In the titration simulations, the barrier height of the biasing potential was set to 7.5 kJ mol -1 for groups modeled with single-site representation, and to 5 kJ mol -1 for groups modeled with the multisite representation.
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The constant pH simulations require a correction potential V MM (λ j ) for each titratable residue j. These correction potentials are the integrals of polynomial fits to the expectation value of ∂V /∂λ λ in reference state simulations at fixed λ values. 15 Thus, after integration, an n th order polynomial fit to ∂V /∂λ λ yields an (n + 1) th order polynomial function that represents V MM (λ). However, in the calculations of our implementation, the fit to ∂V /∂λ λ was used, rather than the V MM (λ). We thus refer to the fitting order as the order of the polynomial fit to the ∂V /∂λ λ .
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We performed the reference simulations as follows: The partial charges in the tripeptide systems were linearly interpolated between λ = -0.1 and λ = 1.1 with a step of 0.05. For His, all three λ-coordinates were changed under the constraint λ 1 + λ 2 + λ 3 = 1. For each set of λ values, called a grid point, we performed an 11 ns MD simulation, in which the ∂V /∂λ j were saved every ps and accumulated. The total charge of the system was kept neutral by simultaneously changing the charge of a single buffer particle. The fitting procedure is described in full detail in the accompanying paper. 15
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Because the sampling of protonation states is tightly coupled to the sampling of side chain dihedral degrees of freedom, we computed the free energy profiles associated with the rotation of the dihedrals in the side chain of the central amino acid in the capped tripeptide systems (Table ) by means of umbrella sampling. As the first step, we performed 20 ns MD simulations with a time-dependent potential on the dihedral angle with a force constant of 418.4 kJ mol -1 rad -2 . The center of this potential was moved from 0 • to 360 • with a rate of 18 • ns -1 . From these simulations, frames with dihedral angles closest to 0 • , 10 • , 20 • , etc.
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were selected as references for the umbrella replicas. The difference between the dihedral angle in the selected frames and the target angle was always below 0.1 • . Then, we performed 36 umbrella sampling simulations of 11 ns with a harmonic restraining potential centered at the reference dihedral angle and a force constant of 418.4 kJ mol -1 rad -2 . We used the WHAM procedure, implemented in GROMACS, to unbias these umbrellas and obtain free energy profiles associated with the full rotation of the dihedral angle.
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To check the validity of the proposed force field modifications, we computed the potential energy profiles for the N-C α -C β -C γ dihedral of aspartic acid with capped residues. The profiles were computed at both quantum mechanical (QM) and molecular mechanical (MM) levels. The QM profiles were computed at the MP2/6-31+G* level of theory using Firefly QC package, which is partially based on the GAMESS (US) source code. The MM profiles were computed for both the original and modified CHARMM36m force fields. The potential energy was computed for N-C α -C β -C γ dihedral angle with 10 • increments. For each dihedral value, the structures were energy-minimized prior to potential energy calculation.
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Buffer particles are used in constant pH MD to maintain the neutrality of the simulated system. Ideally, buffers should not introduce any artifacts due to binding to titratable groups, binding to each other, or penetrating into hydrophobic regions. To prevent such behavior, we optimized the charge range and Lennard-Jones parameters of the buffers. To this end, we performed a series of enhanced sampling simulations with the accelerated weight histogram method (AWH). In one set of simulations with two buffers in the simulation box (BUF 2 , Table ), we quantified the sampling efficiency from the friction metric, as a function of the absolute charge per buffer particle. In these simulations, the charge of one buffer was changed from 0 to +0.8, while simultaneously the charge of the other buffer was changed from 0 to -0.8 in order to maintain neutrality. The CHARMM36m
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Lennard-Jones parameters for sodium were used for the buffers in these simulations. In the other set of simulations, we computed the free energy difference between introducing a neutral buffer in water (BUF 2 ) and inside the hydrophobic region of a POPC bilayer system (MEMB 2 , Table ), for various values of the Lennard-Jones parameters of the buffer. In these simulations, the Lennard-Jones interactions between the buffer and the rest of the system were increased from non-interacting at λ = 0 to fully interacting (λ = 1) in 10 discrete steps.
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where N prot and N deprot are the total number of frames in which the site is protonated and deprotonated, respectively. For titratable sites modeled with the single-site representation, we considered the site protonated if λ is below 0.2, and deprotonated if λ is above 0.8. For sites that are described with the multisite description, we considered a state protonated if the λ associated with the protonated form of the residue is above 0.8, and deprotonated if the λ associated with the deprotonated form of the residue is above 0.8.
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First, we demonstrate that a lack of sampling of the relevant dihedral degrees of freedom in amino acid side chains with the CHARMM36m force field reduces the accuracy of the correction potentials for λ-dynamics. To overcome these convergence problems, we modify the force field by reducing the barriers in the torsion potential and show that this significantly improves the accuracy of the correction potentials and hence the results of constant pH simulations, including pK a estimates, without affecting the protein conformational dynamics.
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After the validation of the modified force field parameters, we show how the buffer particles that maintain the neutrality of the simulation box, have to be parameterized to prevent finite-size effects on proton affinities due to periodicity. Sampling Klimovic and Mobley have shown that calculated hydration free energies of single amino acids depend on the starting conformation. Because a few picoseconds typically suffice to sample bond and angle degrees of freedom in the amino acid, as well as the rotational degrees of freedom of the water molecules, we speculate that their observation implies a lack of sampling in the dihedral degrees of freedom in the amino acid side chain. Therefore,
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We performed 100 ns constant pH MD simulations at pH = pK a of systems ADA, AEA, AKA, AHA, AAA 1 , and AAA 2 (Table ). To enhance the sampling of the λ-coordinate in these systems, we ran the simulations without a barrier in the biasing potential (V bias (λ), Equation 5 in Aho et al. ). The correction potential (V MM (λ), Equation 5 in Aho et al. ) was obtained by fitting a third-order polynomial function to the ∂V /∂λ λ values of the reference trajectories. We will show later that for accurate and reproducible constant pH MD results, a higher-order fit is required. Nevertheless, in spite of its limited accuracy, using the same third-order fit for all system suffices to systematically compare the distributions of the relevant degrees of freedom and assess their convergence.
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In Figure , we show the distributions of the λ-coordinate in five constant pH MD replicas of the ADA system with the original CHARMM36m force field parameters. Distributions of the λ-coordinates in the other systems (AEA, AKA AHA, and AAA 1 and AAA 2 ) are shown in Supporting Information (SI, Figures ). The dissimilarity between the λ-distributions in the replicas (maximum KSS between replicas of 0.29, 0.11, 0.04, and 0.095 for ADA, AEA, AHA, and AAA 1 , respectively) indicates a lack of convergence. In addition, the distributions of the dihedral angles, shown in Figure , c, and d, are also not identical for all replicas. Because there is no barrier from the biasing potential for the λ-coordinate, we conclude that the lack of convergence in λ is due to insufficient sampling of the dihedral degrees of freedom.
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| 22 |
In general, to overcome the lack of sampling, one can increase simulation time, enhance sampling by means of special algorithms such as replica exchange MD, or decrease the barrier of the biasing potential. Since the sampling of λ-coordinates is tightly coupled to the sampling of the dihedral angles, we propose to reduce the barriers of the torsion potentials of the side chains in titratable amino acids. Similar strategies have been also adopted before in constant pH MD simulations. 9, Alternatively, increased sampling of side chain rotamer states has been achieved using pH replica exchange methods. However, these methods are computationally more demanding than performing a single MD simulation and also prevent access to the dynamical properties of the system because of jumps between replicas.
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| 23 |
To improve the sampling of the syn and anti conformations of the carboxyl proton in Glu, Asp, and the C-terminus, Brooks and co-workers reduced the barrier for this rotation by a factor of eight and also scaled the carboxyl oxygen radii by 0.95. 9 In contrast, Grubmüller and co-workers, modified this torsion potential to prevent sampling the anti-conformation altogether. However, according to our analysis (Figure ), there is not only a lack of convergence in the carboxyl dihedral angle but also in the other side chain dihedral angles.
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| 24 |
Reducing the torsional barriers without affecting the overall sampling of the conformational space is possible only if the regions near such barriers are sparsely sampled. We, therefore, analyzed the distributions of the dihedral angles in the side chains of titratable amino acids in the publicly available trajectories of G-protein coupled receptors and of SARS-CoV-2 proteins. The distributions of these dihedral angles, plotted in Figure , reflect the shape of the underlying torsion potentials with maxima coinciding with local minima of the potential profiles. The low density near barriers suggests that these barriers are rather high and might be reduced without affecting the dihedral distributions.
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where n i is the multiplicity of torsion angle i (i.e., the number of minima), with n i = 2 for conjugated bonds and n i = 3 for aliphatic bonds. The parameter i is an empirical coefficient that is optimized such that the barriers are low enough to converge the distribution of φ i , without introducing additional minima on the potential energy surface.
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| 26 |
For each side chain dihedral angle of the titratable amino acids, the coefficient i was optimized in an iterative fashion: After an initial guess, we computed the free energy profiles associated with rotation of the dihedral, as well as five unbiased 100 ns trajectories at pH = pK a with different starting conditions and a biasing potential (V bias (λ), Equation 5 in Aho et al. ) without barrier. Prior to these constant pH MD simulations, we recomputed the correction potential, V MM (λ), by fitting a third-order polynomial to the ∂V /∂λ λ values obtained from thermodynamic integration simulations performed with the current value of i . Free energy profiles were inspected visually for artificial minima, while distributions of both dihedral angles and λ-coordinates were compared between the five unbiased replica runs based on their similarity. The coefficient i was gradually increased until the distributions in the different replicas were sufficiently similar (KSS < 0.03), while at the same time no additional minima appeared in the free energy profiles. Because with the corrected potentials, the Kolmogorov-Smirnov statistics for Asp, Glu, His, and C-terminus are 0.028, 0.015, 0.027, and 0.022, we conclude that the corrections improve the convergence of both the λ and dihedral degrees of freedom in constant pH simulations.
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| 27 |
For many biomolecular force fields, the parameters of the torsion potentials are obtained by fitting suitable periodic functions to energies evaluated at the MP2 level of theory. The parameters for each type of the torsional potential are simultaneously fitted for multiple amino acids. Therefore, the average root mean squared (RMS) difference between the torsional energy at the CHARMM36m level and the MP2 level of theory is of the order of 10 kJ mol -1 . The RMS deviation between the modified and original torsion potentials is at most 8 kJ mol -1 , and the RMS deviation between the ab initio potential at the MP2/6-31+G* level and the N-C α -C β -C γ torsion potential in ASP is reduced from 4 kJ mol -1 for the original CHARMM36m force field to 3.5 kJ mol -1 for modified CHARMM36m force field (Figure ). Therefore, we conclude that with the corrections of the torsion potentials, the modified force field provides an equally good fit to QM potential profiles as the original CHARMM36m force field. We also performed standard MD simulations and simulated five replicas for 100 ns for the two protonation states of the Asp tripeptide in water, using both the original and modified CHARMM36m force field parameters. Without the modifications, the local minima are not consistently sampled in all replicas (Figure ). In contrast, with the corrections, identical distributions of the dihedral angles are obtained also in standard MD simulations.
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Finally, we demonstrate that the modifications do not alter the distributions of the dihedral angles in protein simulations. We performed MD simulations of the 1CVO 1 system both with and without the modifications to the torsion potentials of titratable amino acids, with either (i) all these residues protonated, (ii) all deprotonated, or (iii) all Asp residues deprotonated and all other residues protonated. In Figure we plot the distributions of the dihedral angles for which corrections were introduced. The high similarity between the distributions suggests that the corrections do not lead to the sampling of different dihedral distributions, even if the relative weights of the minima are slightly altered, in particular for the H-O-C-O dihedral. We conclude, that the corrections introduced to facilitate sampling of the dihedral and λ-coordinates, do not significantly alter the protein conformational landscape and can hence be used to perform both normal and constant pH MD simulations.
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| 29 |
However, as shown in Figure , the distributions are identical between replicas, but not uniform, despite the corrections to the torsion potentials. Because both the pH-dependent potential V pH (λ) and the biasing potential are flat by construction at pH = pK a , the deviations must originate from discrepancies between the correction potential V MM (λ) and the underlying free energy profile associated with deprotonation. The correction potential is obtained as a polynomial fit to the ∂V /∂λ λ values from thermodynamic integration simulations. Because linear response (LR) theory predicts a lin-ear dependence between the hydration free energy and the magnitude of a (point) charge, a first-order fit has often been used to obtain the correction potential for constant pH MD. 9,12
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However, even if the change in the charge dominates the free energy of changing the protonation state, hydrogen bond re-arrangements can contribute as well. Because the effects due to such structural rearrangements are neglected in LR theories, we hypothesize that higher-order fits may be necessary for obtaining sufficiently accurate correction potentials.
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To test our hypothesis, we investigated the accuracy of the polynomial fit to the correction potential. In Figure we show the mean error of the correction potential with respect to the computed free energy difference associated with deprotonation, as a function of the fitting order. With an error of 4 kJ mol -1 , the third-order fit, used above to address the convergence issues, does not yield a sufficiently accurate representation of the underlying free energy profile. Increasing the order of the polynomial fit reduces the error and, as shown in Figure , at least seventh order fit is required to provide a uniform distribution of the λ-coordinate for the Asp tripeptide in constant pH MD simulations at pH = pK a .
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| 32 |
Also for carboxyl groups in the side chains of Glu, and in the C-terminus, a polynomial fit to ∂V /∂λ λ of at least seventh order is needed to provide a sufficiently accurate correction potential (Fig. , S11, and S14 in SI). For the imidazole ring of His with three coupled titratable sites, a seventh order fit suffices as well (Fig. ), while for the amino bases in the side chain of Lys and the N-terminus, at least an eight order fit is required (Fig. and S15 in SI). We speculate that the higher-order fit is needed for the latter sites due to the larger change in the charge on the central nitrogen atom from -0.3 e to -0.96 e upon deprotonation. The change in the charge of the carboxylic oxygen from 0.55 e to -0.76 e is smaller, as are the changes on the nitrogen atoms of the imidazole ring of His (-0.36 e to -0.7 e).
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