Abstract
Recent developments in computer processing power lead to new paradigms of how problems in manybody physics and especially polymer physics can be addressed. Parallel processors can be exploited to generate millions of molecular configurations in complex environments at a second, and concomitant freeenergy landscapes can be estimated. Databases that are complete in terms of polymer sequences and architecture form a powerful training basis for crosschecking and verifying machine learningbased models. We employ an exhaustive enumeration of polymer sequence space to benchmark the prediction made by a neural network. In our example, we consider the translocation time of a copolymer through a lipid membrane as a function of its sequence of hydrophilic and hydrophobic units. First, we demonstrate that massively parallel Rosenbluth sampling for all possible sequences of a polymer allows for meaningful dynamic interpretation in terms of the mean first escape times through the membrane. Second, we train a multilayer neural network on logarithmic translocation times and show by the reduction of the training set to a narrow window of translocation times that the neural network develops an internal representation of the physical rules for sequencecontrolled diffusion barriers. Based on the narrow training set, the network result approximates the order of magnitude of translocation times in a window that is several orders of magnitude wider than the training window. We investigate how prediction accuracy depends on the distance of unexplored sequences from the training window.
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Introduction
Polymers are manybody physical objects; in order to describe their equilibrium state and dynamics, it is often required to translate chemical sequence information into freeenergy landscapes in threedimensional space. Rigorous theoretical descriptions can capture only special cases such as homopolymers or multiblock copolymers by following bottom–up approaches that start with interactions on the monomer level or considering the selfsimilarity of selfavoiding walks on the largest scales. The sequence space available by current polymer chemistry^{1,2,3} or in biopolymers exceeds the limits for closed physical descriptions and is not accessible for complete scans by molecular simulation techniques. A new paradigm of datadriven polymer science is increasingly encouraged by parallel sampling methods^{4,5} and the advances in machine learning (ML)^{6,7,8,9,10,11,12,13,14} and has the potential to explore yet undiscovered patterns in sequence–property relationships.
A prominent problem for sequencecontrolled polymers is their transport through lipid membranes and biological barriers, which is linked to a wide field of potential biomedical and biotechnological applications. The translocation time of polymer chains through a narrow nanopore on the scale of one monomer has been described for homopolymers^{15,16} by means of scaling relations and, later on, extended the theory to block copolymers^{17,18}. As soon as local conformation entropy of the polymer comes into play by widening the pore to a finite diameter and length^{19,20}, a general expression as a function of the sequence seems challenging in the moment for both charged and uncharged polymers. The absence of a closed analytic theory for sequencecontrolled translocation meanwhile does not exclude technical applications of nanopores for DNA sequencing^{21,22,23}.
The picture is similar when considering the translocation of a polymer through a lipid membrane by direct penetration of the membrane’s core. Here polymer translocation can be considered as the diffusion of its center of mass along an effective freeenergy landscape determined by the selfassembled membrane environment^{24,25,26}. Translocation of homopolymers through bilayer membranes was recently described theoretically by means of propagators as the solution of Edwards equation^{27} in good agreement with coarse grained simulations^{28}. Simulation results on random copolymers indicate that the main factors for copolymer translocation are their average hydrophobicity as well as their degree of adsorption^{29,30} at the membrane–solvent interfaces^{31}, which shall be reflected in the main modes of their potential of mean force. Experimentally, the passive translocation of synthetic random copolymers into mammalian cells in the absence of cytotoxic effects was discovered^{32} and confirmed recently^{33,34}. A rigorous theoretical description as a function of sequence, however, is missing to date. The lack of theory does meanwhile not exclude the recent progress in finding artificial cellpenetrating peptides and antimicrobial peptides by highthroughput screening^{35,36,37,38} that may even outperform evolutionary highly conserved Tat or Penetratinbased sequences for biomedical application. Wimley et al. found that finetuned differences in shortblock amphiphilic sequences have a significant effect on peptide translocation rates following rules that seems not obvious at the moment^{37}. In turn, sequences leading to optimal points in their biomedical performance can be found in unexpected corners of sequence space that are potentially accessed by sequence–cargo coevolution^{38}. In this work, we make use of a massively parallel sampling of polymer conformations while scanning the full sequence space of a short copolymer and see that the intricacies of polymer sequence–property relationships can be subtle and unexpected already when considering relatively simple environments.
The laws of physics are, however, normally simple by means of requiring a relatively small number of parameters that can be extracted efficiently from highdimensional data by ML methods. Artificial neural networks (NNs) have been successfully applied in the dimensionality reduction from chemical monomer composition of polymers to their material properties, such as glass transition temperature^{39,40,41,42,43}, viscosity^{44}, solvation free energies^{45}, and electronic properties^{13,14} depending on the repeat units. When addressing long polymer chains, the training and test data are fundamentally limited to a fraction of sequences due to the exponential increase of sequence space. Any restriction or bias in the training data has yet undefined consequences for the NN’s projection into unexplored parts of sequence space. Efficient classification and optimization algorithms, such as based on artificial NNs, are in fact “black boxes” and their results therefore need better understanding and explanation. Theoretically, an NN can approximate any continuous mapping given that at least one hidden layer of neurons with sigmoid activation functions is contained^{46,47}. Beyond that, the stacking of nonlinear filters seems to mark a qualitative difference as compared to shallow ML algorithms such that they may develop internal representations of the input information that correspond to a hierarchy of abstraction levels. The distinguished generalization performance makes socalled deep neural nets particularly efficient when confronted with multiple tasks^{48} simultaneously, for instance, in finding quantitative structure–property or structure–activity relationships^{49,50,51,52}. Recent advances in exploiting NNs for physical problems show that they can help to determine the essential order parameters necessary for predicting a mechanical state in future^{10} or classifying a magnetic phases^{8}.
In this work, we have the luxury to access a complete sequencetoproperty map available for training and testing NN algorithms thanks to the graphics processing unit (GPU)accelerated^{4} sampling of random polymer configurations for a given sequence. GPUaccelerated Rosenbluth–Rosenbluth^{53} sampling of a copolymer in an external field modeling a lipid membrane allowed us to generate a significant number of configurations for all possible binary sequences for chain length up to N = 16. Based on this unbiased data, a NN is trained to predict mean first escape times of the polymer through the layer. By systematic selection of a training set, we tested the NN’s performance of projection into unseen parts of the complete sequence space.
The rest of the paper is structured as follows: In section “Results,” we introduce the sequencecomplete sampling data set based on the Rosenbluth–Rosenbluth method for translocation time prediction. By comparison with freeenergy estimates of selfavoiding walks near interfaces, we underline the physical meaning and richness of the results. We also analyze the performance of NN based on translocation time prediction for two different training schemes. In section “Discussion,” we summarize the results. In section “Methods,” we describe the polymer conformation sampling for estimating its translocation time through a membrane as well as the NN model applied.
Results
Rosenbluth–Rosenbluth sampling
Let us consider the inverse mean first escape time 1/τ as a measure for the frequency of translocation of a polymer through the membrane, which is presented in Fig. 1 as a function of the mean hydrophobic fraction along a backbone of N = 12 monomers. Results for all sequences are shown and grouped into point clouds centered at the corresponding ratios N_{T}/N. The point clouds are shaped according to the number, n_{b}, of blocks of H and T species along the sequence in a way that the points on the right hand side of a cloud represent a polymer with a larger number of blocks.
The results in Fig. 1 confirm earlier predictions^{28} that a maximum of translocation frequency is found near a point of balanced hydrophobicity of the polymer as given by a balanced fraction of H and T units N_{T}/N ~ 1/2, in case that the typical block size is in the order of the Kuhn segment of the polymer^{31}.
In Fig. 2, we show the monomer sequences leading to the largest and lowest translocation frequency 1/τ as well as the results for triblock copolymers as a function of chain length for the balanced ratio N_{T}/N = 1/2 of hydrophobic beads. For alternating sequences, the rescaled translocation frequency (see Eq. (7)) remains in the same order of magnitude showing that the polymers are below the adsorption threshold for the given chain lengths. For polymers that are significantly localized at the membrane–solvent interface, one would expect that the desorption to be the ratelimiting process for translocation. Adsorption effect is clearly visible for diblock copolymers showing a nearly exponential decay of translocation frequency as a function of chain length. For diblock copolymers, we expect that the desorption of the hydrophobic block from the membrane is the most significant ratelimiting process, and consequently diblock sequences lead to minimal translocation frequencies. It is important to notice that, for hydrophilic blocks larger than the membrane width, the switch of a hydrophilic end from one solvent side to the opposing solvent does only require a limited number of hydrophilic beads to be in contact with the lipid core at the same time, whereas the escape of the hydrophobic block into the solvent requires all monomers of the block to be displaced into solvent environment. Dynamic barriers such as the steric hindrance of the polymer backbone by lipid tails, is, however, not included in the meanfield environment.
In Fig. 2, it becomes visible that the symmetry of the polymer sequence with respect to hydrophilic ends adds an important factor to the desorption probability, in particular when comparing results for triblock copolymers where the longest chains show a more than one decade larger translocation frequency as compared to diblocks. The difference can be understood qualitatively by estimating the adsorption free energy in the strong segregation limit as
where c is the average number of favored contacts a T monomer finds in the lipid environment (coordination number) and \(\Delta {F}_{{\rm{el}}}={k}_{{\rm{B}}}T\mathrm{ln}\,[{Z}_{{\rm{surf}}}/{Z}_{{\rm{free}}}]\) is an elastic contribution due to the reduction of the partition function from Z_{free} to Z_{surf} upon localization at the surface. The partition sum for a selfavoiding walk takes the form^{58,59,60}
where q is a nonuniversal amplitude that may depend on the particular form of shortrange interactions and μ is the effective coordination number for the given random walk logic and lattice. The exponent γ depends on the topology of the polymer that is either in free solution or attached to a surface. One applies γ ≡ γ_{1} ≈ 0.678^{61,62,63} for strands having one end grafted, and γ ≡ γ_{11} ≈ −0.39^{62,63} for strands having both ends surface attached. The partition sum in free solution scales as \({Z}_{{\rm{free}}} \sim {\mu }^{N}{N}^{{\gamma }_{0}1}\) with γ_{0} ≈ 1.1567^{64,65,66}. Since we further compare only ratios of partition sums for given total chain length, we assume that q and μdependent contributions cancel up to a factor of the order unity.
The probability density to find a symmetric diblock copolymer in bulk solvent as compared to a state adsorbed at an interface as illustrated in Fig. 3 then reads
Now, assuming that the desorption is the ratelimiting process, we write the estimate for the translocation frequency as
In Fig. 2, we show the results for Eq. (3), where T_{0} = 0.123 and c = 19.6 have been adjusted for obtaining leastsquared differences from the diblock RosenbluthRosenbluth sampling (RS) results. The results confirm the dominance of the exponential factor resulting from pair interactions of the hydrophobic block.
The ratio between partition sums for interfaceadsorbed diblocks and triblocks allows to project from diblock to triblock predictions for translocation frequencies,
which is plotted in Fig. 2 for comparison. The resulting upshift catches up to the RS diblock results up to a factor corresponding to a remaining freeenergy difference of 1.4k_{B}T that is missing in Eq. (4). Note that we did not consider finite chain length effects in Eq. (2) in scope of this qualitative comparison.
With this discussion in mind, it is interesting to have a look back to Fig. 1 for understanding surprising features observed in the sequence maps of slightly hydrophilic polymers. By the example of a fraction of 4/12 of hydrophobic monomers, we demonstrate that the polymers comprising the shortest amphiphilic blocks (labelled by “(a)” and “(b)” in Fig. 1) are found in a middle range of translocation frequencies, while triblock copolymers similar as those discussed in Figs. 2 and 3 lead to the largest translocation frequencies. A comparison of the freeenergy profiles shown as an inset in Fig. 1 underlines the interplay between surface adsorption and hydrophobic/hydrophilic balance that leads to the result. Polymers that contain short blocks (“(a)” and “(b)” in Fig. 1) are mainly subject to an effective freeenergy barrier for insertion into the bilayers, which is the ratelimiting factor for translocation. The result reflects the fact that the polymer is effectively hydrophilic and shows negligible surface adsorption effects. Combining Ttype monomers into a larger center block, however, allows for anchoring of the polymer at bilayer–solvent interfaces and thereby effectively reduces the ratelimiting repulsion from the membrane environment. On the other hand, for the diblock copolymers with N_{T}/N = 4/12, adsorption at the bilayer–solvent interface turns over to dominate the freeenergy profiles and leads to the largest escape times found for the given hydrophilic/hydrophobic ratio.
From the comparison between RS results for τ and previous work^{26,28,31} we therefore conclude that the dynamic interpretation of the sampling results is justified.
Machinelearned translocation times
The complete data set generated by GPUaccelerated RS sampling forms a powerful basis for benchmarking the MLbased search for sequences fulfilling given criteria. A network similar to Fig. 4 can be designed in order to predict sequencedetermined properties of a polymer^{39,44,45}. In this work, we stick to the example of logarithmic translocation times log(τ), and refer to a chain length of N = 14 monomers. The total number of sequences excluding the mirrorsymmetric ones is S = 8256. The fraction of sequences within the training set we fix to f_{train} ≡ S_{train}/S = 1/7. The ratio between training to the remaining test set results in 1:6. However, we follow two distinct schemes for the distribution of training sequences within the sequence space: In the uniform scheme, we define equidistant intervals of size 1/f_{train}, along the τsorted sequences (idspace), and select the central sequences within each interval as the training set. In contrast, in the τwindow scheme we select every second sequence within a window S/2 < id ≤ S/2 + 2S_{train}, where “id” is a τsorted unique index in sequence space (section “Methods”). Note that thereby we select sequences within a narrow window in the upper half of translocation times.
In Figs. 5 and 6, we summarize the results of the training, and the performance of the resulting network with respect to the test set. In Fig. 5a, the development of the mean squared error (MSE) between NN and RSbased \(\mathrm{log}\,(\tau )\) values for all test sequences (unseen) is presented. We note a reliable convergence of MSE values for both uniform and τwindow training sets toward a horizontal line indicating that training was stopped early enough for not running into overtraining. In the case of the uniform training set, MSE results typically end up more than one order of magnitude lower as compared to the τwindow training set. The corresponding root mean squared deviation from the expected value typically reduces by a factor of \(\sqrt{2}8 \sim 5\). For the uniform training set, the root mean squared relative deviation from the RSbased \(\mathrm{log}\,(\tau )\) value points to a typical error of 1.0%, whereas for the τwindow training set we observe values of 7.3%.
In Fig. 5b, we show the corresponding mean relative error for the backconverted (not logarithmic) time τ according to
where \(\Delta \mathrm{log}\,(\tau )\) is the absolute difference between the RS and NNbased \(\mathrm{log}\,(\tau )\) values. For the uniform training set, the relative error scatters between −23% and +43% as found for the largest index id_{test} (largest τ), whereas for the fastest polymers 90% of sequences stay within an error of −6% to +15%. For this training set, equivalent to a random selection of sequences, such high accuracy of the network prediction is remarkable when seeing that the RSbased values of τ are spread by a maximum factor of τ_{max}/τ_{min} ~ 2 × 10^{11}. For the τwindow training set, the relative error far away from the training window increases as compared to the uniform set. Nevertheless, as the maximum range of relative errors is found in the interval of −0.87 ≤ Δτ/τ ≤ 1.25 for the largest index id_{test}, we conclude that typically the prediction hits the right order of magnitude for τ despite the fact that we used only the narrow sequence window for training. It is interesting to note that the translocation times of the fastest sequences is typically predicted correctly by a factor of ~3 despite the large distance from the training window. Qualitatively, we expect that, when shifting the τwindow to smaller (larger) values of τ, the prediction accuracy for the smallest τ values will increase (decrease) while accuracy for the largest values of τ will decrease (increase), which is supported by preliminary data (not shown).
Absolute values are not always the main question for the modeled mapping; in some cases it is enough to obtain a decision statement upon the performance of two structures. When comparing two polymer sequences, for instance, we may ask which of those translocates faster. In Fig. 5c, we therefore show the performance of the trained network to give the right answer for this question as a function of sequence id_{test}. For this purpose, we calculated for each NN result τ_{1} for a given test sequence the fraction of all other test sequences leading to an NN output τ_{2} that holds the same relation τ_{1} > τ_{2} or τ_{1} < τ_{2} as the corresponding pair of RS sampling results. In case of uniform training, \(98.{7}_{1.6}^{+1.1} \%\) of other sequences are correctly attributed as slower or faster (with a confidence of 90%), and for the τwindow training set \(96.{8}_{4.5}^{+2.8} \%\) of pairs are correctly labeled. For the τwindow training set, the performance far away from the training window is reduced, in particular for sequences with a lower id_{test} index. However, the average fraction of correct decisions does not drop below 94.3% for the selected bin size.
By Fig. 5, we therefore demonstrated that a quantitative prediction of translocation times is possible by the applied ML model, and the accuracy depends crucially on the distribution of training sequences.
In Fig. 6, we outline more details of the training result by showing the predicted value of \(\mathrm{log}\,(\tau )\) for the whole test sets of uniform and τwindow in Figs. 6a and 6b, respectively. The monotony of the predicted data points for both training sets follows the base data line despite the scattering of the data as discussed for Fig. 5. In particular, for the τwindow training set, we emphasize that the order of translocation times is predicted correctly for the fastest sequences although the training set covers only a narrow window within the slower half of sequences.
Another interesting observation is the prediction of steplike features in translocation time (arrows in Fig. 6b) as function of id_{test} that are reproduced throughout the test set although located outside of the τwindow training range. Thus even the relatively simple network seems capable of finding a generic rule that links sequence and translocation time and thereby expresses the rather rich result based on Eqs. (7) and (8) without knowledge of conformation entropy nor the escape times. In view of the generalization performance observed for the τwindow training set, it therefore seems that the network developed an implicit internal representation approximating the mathematical rules linking copolymer sequence and translocation time.
Discussion
We apply a massively parallel sampling of the conformations of amphiphilic copolymers by means of selfavoiding random walks within a given density field representing a model for amphiphilic bilayer membranes. We estimated the freeenergy profiles of the polymers composed of hydrophilic (H) and hydrophobic beads (T) with respect to distance from the membrane as a reaction coordinate. We calculated the mean first escape time τ as a measure for polymer translocation time through the model membrane all 2^{N} binary sequences up to chain length N ≤ 16. Our results confirm that polymer translocation is controlled by a balance of the overall hydrophobicity of the polymer and is inhibited by adsorption at the bilayer–solvent interfaces^{26,27,28,31}, which is consistent with the picture for small solutes^{67} and larger solid objects such as carbon nanotubes^{68}.
Amphiphilic polymers at a balanced hydrophobicity show the smallest translocation times when the sequence exposes small repeating amphiphilic features, while longest waiting times are associated with a diblock structure of the whole chain. The different translocation rates between diblock and triblock copolymers as well as their chainlength dependence can be explained qualitatively when comparing adsorptionfree energies at the bilayer–solvent interface involving surfacecritical exponents. The relatively weak dependence of the translocation time of balanced hydrophobicity smallblock alternating copolymers from chain length indicates that local amphiphilic features are only weakly interacting with the bilayer–solvent interfaces and the copolymer effectively resembles a homopolymer chain for which the membrane is energetically transparent. Chainlength dependence in this case is expected to increase when effective monomer association constants are stronger than in the present model. When considering slightly hydrophilic backbones, larger hydrophobic blocks start to become more prominent in sequences leading to smallest translocation times as they promote the association of the netrepulsive backbone with the hydrophobic membrane core.
The extensive database generated by RS sampling has been used to feed a multilayer artificial NN with four hidden layers in order to explore the capability of socalled deep learning approaches for finding a general rule of how copolymer sequence translates into translocation times through biological barriers. The aim of this work is to test the meaningful interpretation of the “dirty work” of NNs provided by a complete data set of polymer sequences. We demonstrate that, even by using a low fraction 1/7 of uniformly selected training examples as compared to the total number 2^{N} of binary sequences for N = 14, the NN achieves a root mean squared relative deviation in the order of 1% for the logarithmic mean first escape time \(\mathrm{log}\,(\tau ).\) In order to test the generalization performance of the network, we implemented a second training scheme, where training examples have been selected from a narrow window of sequences with respect to translocation times τ covering a factor of ≈30 between maximum and minimum translocation times contained in the training set. In this case, the network approximates the order of magnitude of the test data set covering a window being more than nine orders of magnitude wider. We conclude that the NN developed an internal representation of the mathematical rules linking sequence and translocation times, which involve a precise estimate of ratelimiting energy barriers. The network thereby encodes a complex interplay between polymer net hydrophobicity and sequencedependent adsorption at the bilayer–solvent interfaces that to date can be treated in a theoretically closed form only for special cases as it involves the sequencedependent polymer conformation entropy and solving the diffusion problem in inhomogeneous freeenergy landscapes. Our results indicate a systematic decrease of prediction accuracy when moving into unexplored corners of sequence space and challenge future investigation on the relation between training data bias and prediction accuracy.
Methods
Rosenbluth–Rosenbluth sampling
We consider the diffusive transport of a polymer through a lipid membrane resembling a homogeneous oil slab as shown in Fig. 7. In particular, we are interested in mean first escape time of a polymer through the membrane as a function of length, N, sequence of hydrophilic head (H) and hydrophobic tail monomers (T). Coarse grained polymers are embedded into an external concentration field that represents bilayer membrane on a meanfield level composed of an hydrophilic region (H) and a hydrophobic core (T), as well as solvent (S). The hydrophobic core has a thickness of six lattice units.
Monomers are represented as singlecell occupations on a simple cubic lattice, and bond vectors are taken from a set of 26 vectors with lengths of 1, \(\sqrt{2}\), and \(\sqrt{3}\) lattice units. Double occupancy of lattice sites is forbidden, and the monomers have excluded volume. This set of static rules corresponds to those of Shaffer’s Bond Fluctuation Model^{54}.
Between hydrophilic sites (H and S), and hydrophobic sites (T), we implement shortrange repulsive interactions. We write the internal energies of H and T monomers of the polymer as
where \({c}_{x}(\overrightarrow{r})\) are the number of lattice occupancies by species x on the 26 nearest neighbor sites^{55}. In order to keep the model simple, we use only a single interaction parameter defined as ϵ = 0.1k_{B}T with k_{B} being Boltzmann’s constant and T the absolute temperature. For the enumeration of c_{x}, the occupancy of the lattice by a given external concentration field (Fig. 7) is counted, and monomer–monomer contacts are taken into account in a way that contacts with the external field are screened by surrounding monomers. Thereby solventinduced effects on polymer conformations are represented by the model.
For a given amphiphilic sequence, we aim to calculate the mean first escape time of a polymer between a repulsive boundary at z = −a and an absorbing boundary at z = +a, (Fig. 7)^{56},
where D is the diffusion constant of the polymer and p(z) is the probability distribution to find the center of mass of the polymer at a given distance, z, from the bilayer’s midplane. We define a = 22 lattice units, and D = 1(lattice unit)^{2}, such that the dimensionless number of τ does not include the chainlength dependence of diffusion time.
The probability distribution p(z) is calculated by generating M polymer conformations \(\overrightarrow{R}=({\overrightarrow{r}}_{1},{\overrightarrow{r}}_{2},\ldots ,{\overrightarrow{r}}_{N})\) according to the RS scheme^{53}. For each conformation \(\overrightarrow{R}\), the contact energy \(U(\overrightarrow{R})\) is calculated according to \(U(\overrightarrow{R})=\mathop{\sum }\nolimits_{i = 1}^{N}{U}_{X}({\overrightarrow{r}}_{i})\) in units of k_{B}T according to Eq. (6) depending on the species X of the monomer X = H or X = T. The center of mass \(\bar{z}(\overrightarrow{R})=(1/N)\mathop{\sum }\nolimits_{i = 1}^{N}{\overrightarrow{r}}_{i}{\overrightarrow{e}}_{z}\) is evaluated with \({\overrightarrow{e}}_{z}\) being the lattice unit vector along the membrane’s normal direction. The distribution p(z) is then written as
where the condition below the sum illustrates that only those conformations contribute whose center of mass is found within a grid distance \((z1/2)\;<\;\bar{z}\le (z+1/2)\) from z, and β ≡ 1/(k_{B}T). In Eq. (8), W_{i} is the Rosenbluth weight of the ith conformation.
For a given sequence of H and T monomers in a polymer backbone, we calculate the mean first escape time according to Eq. (7) based on the generation of M = 1.5 × 10^{7} RSgenerated chains at uniformly distributed random positions within a periodic lattice of 64 × 64 × 64 lattice sites. The algorithm is implemented for GPUs^{4}. In order to analyze how the mean first escape time depends on the amphiphilic sequence of the polymer, we perform the procedure for all 2^{N} sequences for various degrees of polymerization N ≤ 16.
Artificial NN
We employ a fully connected NN involving \({\tan}{\rm{h}}\)activation as sketched in Fig. 4. The network is composed by 2 hidden layers with 64 nodes each followed by 2 hidden layers with 32 nodes each. The input layer corresponds to a vector of values 0 and 1 representing the considered amphiphilic sequence of hydrophobic (0) and hydrophilic (1) monomers. The output layer consists of one neuron whose output is compared to the RSbased τ value for this sequence. The total network depth is n = 5, where only the output activation, \({\tan}{\rm{h}} [{\sum \nolimits_{i = 1}^{32}}({w_{n,i}}{h}_{n1,i}+b)]\), includes a bias, b. Since absolute values of τ spread over several orders of magnitude, we perform the training with respect to its logarithm. The RSbased values of \(\mathrm{log}\,(\tau )\) are further linearly normalized and centralized into an interval [−0.9, 0.9] by defining \(I({\log}\,(\tau ))=1.8\times [({\log}\,(\tau ){\log}\,({\tau }_{{\rm{min}}}))/({\log}\,({\tau }_{{\rm{max}}}){\log}\,({\tau }_{{\rm{min}}}))\frac{1}{2}]\) in order to be conveniently expressible by the \({\tan}{\rm{h}}\)activation output. NNbased estimates for \(\mathrm{log}\,(\tau )\) are obtained by the back projection, I^{−1}, of output neuron activations.
All weights are initialized with uniform random numbers in an interval [−0.3, 0.3]. The feedforward (ff) backpropagation (bp)^{57} algorithm is employed for training. Error bp is performed after each ff cycle for a randomly selected sequence taken from the training set (stochastic gradient descent). The squared difference between the resulting activation of the output neuron and the RSbased \(I(\mathrm{log}\,(\tau ))\) value is used as the cost function for weight and bias adjustment. We set the initial training rate to η = 0.02, which gets reduced by a factor of (1/1.3) every 10^{3} epochs in order to avoid frustration or early overtraining effects. One epoch is defined as the average number of ffbp cycles per sequence–τ pair. We set the total number of epochs to 10^{4}.
For each sequence, we define an unique integer identifier, 1 ≤ id ≤S, that is sorted according to the RSbased τ value. A lower id means a lower τ. The whole of S sequences is divided into a training set of size S_{train} and a test set of the size S_{test} = S − S_{train}. For the test set, we define a unique identifier id_{test} for each sequence that is the analog to id for the total sequence space. The index id_{test} labels sequences that are unseen by the network during training.
Data availability
The data used in the paper are available from the authors upon request.
Code availability
The source code of the programs used in this paper is available from the authors upon request.
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Acknowledgements
M.W. and V.A.B. gratefully thank the EU’s Marie Curie Actions under European Union 7th Framework Programme (FP7), Initial Training Network SNAL Grant No. 608184. Y.G. acknowledges funding from the NSFC grant number 11804151 and FRFCU 14380017. V.A.B. acknowledges financial assistance from the Ministerio de Ciencia, Innovación y Universidades of the Spanish Government (CTQ201784998P).
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M.W. and V.A.B. conceived the idea, Y.G. designed and performed Rosenbluth Sampling method calculations, M.W. designed artificial neural network and performed neural network and Rosenbluth calculations. M.W. wrote the manuscript with contribution of all authors. All the authors participated in discussions and contributed materially in finalizing the paper.
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Werner, M., Guo, Y. & Baulin, V.A. Neural network learns physical rules for copolymer translocation through amphiphilic barriers. npj Comput Mater 6, 72 (2020). https://doi.org/10.1038/s4152402003185
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DOI: https://doi.org/10.1038/s4152402003185
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