Abstract
Many complex processes, from protein folding to neuronal network dynamics, can be described as stochastic exploration of a highdimensional energy landscape. Although efficient algorithms for cluster detection in highdimensional spaces have been developed over the last two decades, considerably less is known about the reliable inference of state transition dynamics in such settings. Here we introduce a flexible and robust numerical framework to infer Markovian transition networks directly from timeindependent data sampled from stationary equilibrium distributions. We demonstrate the practical potential of the inference scheme by reconstructing the network dynamics for several proteinfolding transitions, generegulatory network motifs, and HIV evolution pathways. The predicted network topologies and relative transition time scales agree well with direct estimates from timedependent molecular dynamics data, stochastic simulations, and phylogenetic trees, respectively. Owing to its generic structure, the framework introduced here will be applicable to highthroughput RNA and proteinsequencing datasets, and future cryoelectron microscopy (cryoEM) data.
Introduction
Energy landscapes encapsulate the effective dynamics of a wide variety of physical, biological, and chemical systems^{1,2}. Wellknown examples include a myriad of biophysical processes^{3,4,5,6,7}, multiphase systems^{2}, thermally activated hopping in optical traps^{8,9}, chemical reactions^{1,10}, brain neuronal expression^{11}, cellular development^{12,13,14,15,16}, and social networks^{17}. Energetic concepts have also been connected to machine learning^{18} and to viral fitness landscapes, where pathways with the lowest energy barriers may explain typical mutational evolutionary trajectories of viruses between fitness peaks^{19,20}. Recent advances in experimental techniques including cryoelectron microscopy (cryoEM)^{3,21,22} and singlecell RNAsequencing^{23}, as well as new online social interaction datasets^{24}, are producing an unprecedented wealth of highdimensional instantaneous snapshots of biophysical and social systems. Although much progress has been made in dimensionality reduction^{25,26,27} and the reconstruction of effective energy landscapes in these settings^{3,13,16,17,28}, the problem of inferring dynamical information such as proteinfolding or mutation pathways and rates from instantaneous ensemble data remains a major challenge.
To address this practically important question, we introduce here an integrated computational framework for identifying metastable states on reconstructed highdimensional energy landscapes and for predicting the relative mean first passage times (MFPTs) between those states, without requiring explicitly timedependent data. Our inference scheme employs an analytic representation of the data based on a Gaussian mixture model (GMM)^{29} to enable efficient identification of minimumenergy transition pathways^{30,31,32}. We show how the estimation of transition networks can be optimized by reducing the dimension of a highdimensional landscape while preserving its topology. Our algorithm utilizes experimentally validated analytical results^{8,9} for transition rates^{1,33,34,35}. Thus, it is applicable whenever the time evolution of the underlying system can be approximated by a Fokker–Plancktype Markovian dynamics, as is the case for a wide range of physical, chemical, and biological processes^{1,34}.
Specifically, we illustrate the practical potential by inferring proteinfolding transitions, stateswitching in generegulatory networks, and HIV evolution pathways. Current standard methods for coarsegraining the conformational dynamics of biophysical structures^{36,37} typically estimate Markovian transition rates from timedependent trajectory data in largescale molecular dynamics (MD) simulations^{38,39}. By contrast, we show here that proteinfolding pathways and rates can be recovered without explicit knowledge of the timedependent trajectories, provided the system is sufficiently ergodic and equilibrium distributions are sampled accurately. Furthermore, we show that the dynamics of stateswitching or phenotypeswitching in generegulatory networks^{40} can be inferred directly from static snapshots of protein abundances in regimes where deterministic modeling only captures a single steady state^{41,42}. The agreement of our inferred results with two separate sets of timedependent measurements suggests that the inference of complex transition networks via reconstructed energy landscapes can provide a viable and often more efficient alternative to traditional timeseries estimates, particularly as new experimental techniques will offer unprecedented access to highdimensional ensemble data.
Results
Minimumenergypath network reconstruction
The equilibrium distribution \(p\left({\bf{x}}\right)\) of a particle diffusing over a potential energy landscape \(E({\bf{x}})\) is the Boltzmann distribution \(p({\bf{x}})=\exp \left[E({\bf{x}})/{k}_{\text{B}}T\right]/Z,\) where \({k}_{\text{B}}\) is the Boltzmann constant, \(T\) is the temperature, and \(Z\) is a normalization constant. Given the probability density function (PDF) \(p({\bf{x}})\), the effective energy can be inferred from
where \({p}_{\max }\) is the maximum value of the PDF, included to fix the minimum energy at zero. Our goal is to estimate the MFPTs between minima on the landscape using only sampled data. We divide this task into three steps, as illustrated in Fig. 1 for test data (Supplementary Methods). In the first step, we approximate the empirical PDF by using the expectation maximization algorithm to fit a GMM in a space of sufficiently large dimension \(d\) (Methods and Fig. 1a). Mixtures with a bounded number of components can be recovered in time polynomial in both the dimension \(d\) and the required accuracy^{43}. The resulting GMM yields an analytical expression for \(E({\bf{x}})\) via Eq. (1).
In the second step, the inferred energy landscape \(E({\bf{x}})\) is reduced to a minimumenergypath (MEP) network whose nodes (states) are the minima of \(E({\bf{x}})\) (Fig. 1b top). Each edge represents an MEP that connects two adjacent minima and passes through an intermediate saddle point (Fig. 1b). The MEPs are found using the nudged elastic band (NEB) algorithm^{30,31}, which discretizes paths with a series of beadspring segments (Supplementary Methods).
Markov state model
Given the MEP network, the final step is to infer the rates for transitioning from a minimum \(\alpha\) to an adjacent minimum \(\beta\). Assuming overdamped Brownian dynamics, the directed transition \(\alpha \to \beta\) can be characterized by the generalized Kramers transition rate^{1}
where \(\gamma\) is the effective friction, \({E}_{\text{b}}\) is the energy difference between the saddle point \(S\) on the MEP (over the energy barrier) and the minimum \(\alpha\), \({\omega }_{i}^{\alpha }\) are the stable angular frequencies at the minimum \(\alpha\), and \({\omega }_{i}^{S}\) and \({\omega }_{\text{b}}\) are the stable and unstable angular frequencies at the saddle, respectively. Equation (2) assumes isotropic friction but can be generalized to a tensorial form^{1} if anisotropies are relevant. In most practical applications, the error from assuming \(\gamma\) to be isotropic is likely negligible compared with other experimental noise sources. In principle, Eq. (2) can be refined further by including quartic (or higher) corrections to the prefactor \({\omega }_{\text{b}}/\gamma\) to account for details of the saddle shape^{1}. Such corrections can be significant for GMMs (Supplementary Methods).
Each edge \((\alpha \beta )\) has two weights, \({k}_{\alpha \beta }\) and \({k}_{\beta \alpha }\), assigned to it. The rate matrix \(({k}_{\alpha \beta })\) completely specifies the Markov state model (MSM) on the network. Solving the MSM yields the matrix of pairwise MFPTs between states (Fig. 1c and Methods). In a simple twostate system, the MFPTs are determined up to a time scale by detailed balance, but for three or more states the influence of landscape topography and the associated state network topology (Methods) can lead to interesting hierarchical ordering of passage times. Identifying these hierarchies and ways to manipulate them is the key to controlling proteinfolding or viral evolution pathways.
Topologypreserving dimensionality reduction
To ensure that the inference protocol can be efficiently applied to larger systems with a highdimensional energy landscape, we derive a general method for reducing the dimension \(D\) of an energy landscape while preserving its topology. A PDF with \(C\) wellseparated Gaussians in \(D\) dimensions can be projected onto the \(d=C1\) dimensional hyperplane spanning the Gaussian means using principal component analysis (PCA); projecting onto a hyperplane of dimension \(d\, \, 1\) risks losing information about the relative positions of the Gaussian means and, in general, does not allow a correct recovery of the MFPTs (Supplementary Methods). In practice, it suffices to choose \(C\) to be larger than the number of energy minima if their number is not known in advance.
To preserve the topology under such a transformation—which is essential for the correct preservation of energy barriers and MEPs in the reduceddimensional space—one needs to rescale GMM components in the lowdimensional space depending on the covariances of the Gaussians in the \(Dd\) neglected dimensions (Fig. 1c). Explicitly, one finds that within the subspace spanned by the retained principal components (Supplementary Methods)
as long as \(p\) satisfies certain minimally restrictive conditions (Supplementary Methods). Here, \({{\bf{U}}}_{d}\) denotes the first \(d=C1\) columns of the matrix of sorted eigenvectors \({\bf{U}}\) of the covariance matrix of the Gaussian means, and \({\phi }_{i}\), \({p}_{i}^{d}\) and \({\mathbf{\Sigma} }_{i}\) are the mixing components, reduceddimensional PDF, and the covariance matrix of each individual Gaussian in the mixture, respectively (Supplementary Methods). Neglecting the determinant scale factors in Eq. (3), as is often done when GMM models are fitted to PCAprojected data, leads to inaccurate MFPT estimates (Fig. 1c, bottom). It is noteworthy that Eq. (3) does not represent inversion of the transformation performed on the data by PCA, unless all \(D\) dimensions are retained; if some dimensions are neglected, Eq. (3) represents a rescaling of the marginal distribution in the retained dimensions to reconstruct the PDF in the original dimension. In other words, the transition rates are best recovered from the conditional—not marginal—distributions, which are given by Eq. (3) up to a constant factor that does not affect energy differences.
Dimensionality reduction can substantially improve the efficiency of the NEB algorithm step as follows: when the MEPs in the reduced \(d\)dimensional space have been computed, the identified minima and saddles can be transformed back into the original data dimension \(D\) to calculate the Hessian matrices at these points, allowing Kramers’ rates to be calculated as usual (Fig. 1c and Supplementary Methods). Alternatively, in specific situations where the MEPs lie outside the hyperplane spanning the means (Supplementary Methods), the MEP in the reduced \(d\)dimensional space can be transformed back to the \(D\)dimensional space and can be used as an initial condition in that space, significantly reducing computational cost. These results present a step towards a general protocol for identifying reaction coordinates or collective variables for projection of a highdimensional landscape onto a reduced space, while quantitatively preserving the topology of the landscape.
Protein folding
To illustrate the vast practical potential of the above scheme, we demonstrate the successful recovery of several proteinfolding pathways, using data from previous largescale MD simulations^{38}. The protein trajectories, consisting of the timedependent coordinates of the alpha carbon backbone, were preprocessed, subsampled by a factor of 5, treated as a set of static equilibrium measurements, and reduced in dimension before fitting a GMM (Methods). As is typical for highdimensional parameter estimation with few structural assumptions, the fitting error due to a finite sample size \(n\) in \(d\) dimensions scales approximately as \(\sqrt{d/n}\) (Supplementary Methods); see refs. ^{44,45,46} for advanced techniques tackling samplesize limitations. Here, \(d\,<\,10\) so the sample size \(n \sim 1{0}^{5}\) suffices for effective recovery; indeed, our results were found to be robust for trajectories further subsampled by up to a factor of 25, leaving around 500 samples per Gaussian (Supplementary Fig. 3).
For each of the four analyzed proteins Villin, BBA, NTL9, and WW, the reconstructed energy landscapes reveal multiple states including a clear global minimum corresponding to the folded state (Fig. 2a, b). To estimate MFPTs, we determined the effective friction \(\gamma\) in Eq. (2) for each protein from the condition that the line of best fit through the predicted vs. measured MFPTs has unit gradient. Although not usually known, \(\gamma\) could in principle be calculated by incorporating timedependent information from MD simulations or experimental data. Our MFPT predictions agree well with direct estimates (Supplementary Methods) from the timedependent MD trajectories (Fig. 2c). Detailed analysis confirms that the MFPT estimates are robust under variations of the number of Gaussians used in the mixture (Supplementary Fig. 1). Also, the estimated MEPs are in good agreement with the typical transition paths observed in the MD trajectories (Supplementary Fig. 2).
Generegulatory networks
Next, we demonstrate the ability of our protocol to infer stateswitching pathways in multistable generegulatory networks. Using a Gillespie stochastic simulation algorithm (SSA; Methods), we simulated three repressilatortype generegulatory network motifs^{47} with selfactivation. Gene network motifs with features such as these have been studied extensively in recent years, owing to their ability to exhibit precise oscillations^{48} and to their possible importance in the determination of multiple cell fates^{49} in the appropriate parameter regimes, although the role of noise in such networks is not well understood. In our simulated gene networks, each gene encodes a protein that activates the expression of its associated gene and represses another, with \(D=2\), 3, and 4 dimensions at low molecule numbers (Fig. 3a and Supplementary Methods). In each case, parameters were chosen to preclude oscillatory dynamics (Fig. 3a). The energy landscapes reconstructed from the simulation datasets in protein moleculenumber space (with timedependence removed) revealed multiple metastable states for each network (Fig. 3b and Supplementary Fig. 5). Broadly, we found each state to correspond to a mixture of low and high abundances for each separate protein, with the two most common states in \(D=4\) dimensions consisting of two abundant and two depleted proteins (Fig. 3b). In agreement with previous studies^{41,42}, the identified metastable states were not recovered from deterministic simulations of the governing ordinary differential equations (Supplementary Methods), but could only be identified directly from the stochastic data (Fig. 3a, b). We determined the effective friction \(\gamma\) in Eq. (2) for each \(D\) as in the protein example. The predicted MFPTs and MEPs between each metastable state were found to be accurate in comparison with timedependent measurements (Fig. 3c and Supplementary Fig. 5b) and were robust to measurement noise typically encountered in singlecell sequencing (Supplementary Fig. 6). Our framework also correctly predicted MFPTs for a 5D asymmetric gene network (Supplementary Fig. 7). These results demonstrate the utility of our protocol for generegulatory network datasets and, more generally, energy landscapes in discrete spaces.
Viral evolution
As a final proofofconcept application, we demonstrate that our inference scheme recovers the expected evolution pathways between HIV sequences as well as the key features of a distancebased phylogenetic tree (Fig. 4). To this end, we reconstructed an effective energy landscape from publicly available HIV sequences sampled longitudinally at several points in time from multiple patients^{50}, assuming that the frequency of an observed genotype is proportional to its probability of fixation and that the highdimensional discrete sequence space can be projected onto a continuous reduceddimensional phenotype space (Fig. 4a and Supplementary Methods). First, a Gaussian was fit to each patient and then combined in a GMM with equal weights, to avoid bias in the fitness landscape towards sequences infecting any specific patient (Supplementary Methods). Thereafter, we applied our inference protocol to reconstruct the effective energy landscape, transition network (Fig. 4b), and disconnectivity graph (Fig. 4c), where each state is associated to a separate patient. As expected, states corresponding to patients infected with different HIV subtypes are not connected by MEPs (Fig. 4a, b). The disconnectivity graph reproduces the key features of a coarsegrained patientlevel representation of the phylogenetic tree (Fig. 4c). Using our inference scheme, vertical evolution in the tree can be tracked along the MEPs in a reduceddimensional sequence space (Fig. 4b). The energy barriers, represented by the lengths of the vertical lines in the disconnectivity graph (Fig. 4c), provide an estimate for the relative likelihood of evolution to fixation via point mutations between fitness peaks (energy minima). If mutation rates are known, the MEPs can also be used to estimate the time for evolution to fixation from one fitness peak to another^{51}.
Discussion
Finding the appropriate number of collective macrovariables to describe an energy landscape is a generic problem relevant to many fields. For example, although some proteins can be described through effective onedimensional reaction coordinates^{5,7,52,53}, the accurate description of their diffusive dynamics over the full microscopic energy landscape requires many degrees of freedom^{54,55}. Whenever dynamics are inherently highdimensional, topologypreserving dimensionality reduction can enable a much faster search of the energy landscape for minima and MEPs. In practice, data dimension is often reduced with PCA or similar methods before constructing an energy landscape^{55,56,57,58,59,60,61,62}. The extent to which commonly used dimensionality reduction techniques alter MEP network topology or quantitatively preserve energy barriers is not well understood. Equation (3) suggests that reducing dimensions using PCA should not introduce significant errors if the variance of the landscape around each state (energy minimum) in the neglected dimensions is similar. For instance, we found that the proteinfolding data could be reduced to five dimensions while maintaining accuracy (Supplementary Fig. 1), although additional higher energy states may become evident in higher dimensions. As an alternative to using Eq. (2) in the last stage of our approach, a method such as maximum caliber^{63,64,65}, which does not take the derivatives of landscape topology into account, could be supplied with the sizes of the energy barriers and used to infer MFPTs. However, we found that owing to the dependence of the MFPTs on the prefactors in Eq. (2) for different transitions, this technique could not recover all transition rates accurately for either proteins or generegulatory networks (Supplementary Fig. 4). Overall, our theoretical results demonstrate the benefits of combining an analytical PDF with a linear dimensionality reduction technique so that the neglected dimensions can be accounted for explicitly.
Rapidly advancing imaging techniques, such as cryoEM, will allow many snapshots of biophysical structures to be taken at the atomic level in the near future^{3,21,22,28,66,67}. A biologically and biophysically important task will be to infer dynamical information from such instantaneous static ensemble measurements. The proteinfolding example in Fig. 2 suggests that the framework introduced here can help overcome this major challenge; in principle, the framework requires only the pairwise distances between recognizable features of the protein as input (here we used the carbon alpha coordinates). Another promising area of future application is the analysis of singlecell RNAsequencing data quantifying the expression within individual cells^{23}. Related to this application, Fig. 3 demonstrates that our protocol recovers stateswitching pathways in multistable generegulatory networks, which are thought to underlie cellfate decisions. These results are most relevant in lowmoleculenumber regimes, in which noise is known to be an important factor^{68}. In relevant recent work, an effective nonparametric energy landscape of singlecell expression snapshots was inferred using the Laplacian of a knearest neighbor graph on the data, allowing lineage information to be derived via a Markov chain^{15}. The GMMbased framework here provides a complementary parametric approach for reconstructing faithful lowdimensional transition state dynamics from such highdimensional data.
Furthermore, the proofofconcept results in Fig. 4 suggest that our inference scheme for Markovian network dynamics can be useful for studying viral and bacterial evolution, which are often modeled as movements through a series of DNA or protein sequences^{69}. The fitness landscape of an organism in sequence space is analogous to the negative of an effective energy landscape. The process of fixation by a succession of mutants in a population, whereby each mutant replaces the previous lineage as the population’s most recent common ancestor, has been modeled as a Markov process^{70}. Successive sweeps to fixation have been observed in longterm evolution experiments, promising groundbreaking data for future analysis as wholegenome sequencing technologies improve^{71}.
The inference protocol opens the possibility to analyze previously intractable multiphase systems: many highdimensional physical, chemical, and other stochastic processes can be described by a Fokker–Planck dynamics^{1}, with phase equilibria corresponding to maxima of the stationary distribution. By taking nearsimultaneous measurements of many subsystems within a large multistable Fokker–Planck system, the above scheme allows the inference of coexisting equilibria and transition rates between them. Other possible applications may include neuronal expression^{11} and social networks^{17,24}, which have been described in terms of effective energy landscapes.
Although we focused here on normal whitenoise diffusive behavior, as is typical of proteinfolding dynamics, the above ideas can in principle be generalized to other classes of stochastic exploration processes. Such extensions will require replacing Eq. (2) through suitable generalized rate formulas, as have been derived for correlated noise^{1}. Conversely, the present framework provides a means to test for diffusive dynamics: if the MFPTs of an observed system differ markedly from those inferred by the above protocol, then either important degrees of freedom have not been measured, the system is out of equilibrium on measurement time scales, or the system does not have Brownian transition statistics, necessitating further careful investigation of its time dependence.
By construction, the above framework is applicable to systems whose steadystate dynamics is approximately Markovian and can be described by a Fokker–Plancktype dynamics. This broad class includes thermal equilibrium systems as well as nonequilibrium systems that can be approximated by effective equilibrium theories^{72,73}. However, such approximations can become inaccurate if probabilistic nonequilibrium fluxes dominate the system dynamics^{74}. For example, reconstructing dynamical geneexpression information from static snapshots is sometimes possible in the presence of oscillatory dynamics caused by processes such as the cell cycle, but can fail for gene networks with large oscillations that are not orthogonal to the processes of interest^{15}. Adapting the above protocol to reconstruct the dynamics in the latter case, and of farfromequilibrium systems in general, will require incorporating more sophisticated theories that include timeresolved information^{75,76,77,78} and improved expressions for nonequilibrium transition rates^{79}, and account for probabilistic fluxes^{80}.
To conclude, the conformational dynamics of biophysical structures such as viruses and proteins, and the stateswitching dynamics of noisy generegulatory networks, are characterized by their metastable states and associated transition networks, and can often be captured through Markovian models. Current experimental techniques, such as cryoEM or RNAsequencing, provide limited dynamical information. In these cases, transition networks must be inferred from static snapshots. Here we have introduced and tested a numerical framework for inferring Markovian state transition networks via reconstructed energy landscapes from highdimensional static data. The successful application to proteinfolding, generegulatory network, and viral evolution pathways illustrates that highdimensional energy landscapes can be reduced in dimension without losing relevant topological information. In general, the inference scheme presented here is applicable whenever the dynamics of a highdimensional physical, biological, or social system can be approximated by diffusion in an effective energy landscape.
Methods
Population landscapes
A GMM was used to represent the PDF, or population landscape, of samples. The PDF at position \({\bf{x}}\) of a GMM with \(C\) mixture components in \(d\) dimensions is
where \({\phi }_{i}\) are the weights of each component, \({\boldsymbol{\mu} }_{i}\) are the means, and \({\boldsymbol{\Sigma} }_{i}\) are the covariance matrices. More details on GMMs and how they were fit to data are given in the Supplementary Methods.
Mean first passage times
We form a discretestate continuoustime Markov chain on states given by the minima of the energy landscape. For a pair of states \(\alpha\) and \(\beta\) directly connected by a minimumenergy pathway via a saddle, we approximate the transition rate \(\alpha \to \beta\) by the Kramers rate \({k}_{\alpha \beta }\) in Eq. (2), whereas if \(\alpha\) and \(\beta\) are not directly connected we set \({k}_{\alpha \beta }=0\). Given these rates, the Markov chain has generator matrix \({M}_{\alpha \beta }\) where \({M}_{\alpha \beta }={k}_{\alpha \beta }\) for \(\alpha \,\ne\, \beta\) and \({M}_{\alpha \alpha }={\sum }_{\beta :\beta \ne \alpha }{k}_{\alpha \beta }\). Then the matrix \({\tau }_{\alpha \beta }\) of MFPTs (hitting times) for transitions \(\alpha \to \beta\) satisfies
Protein data preprocessing
Proteinfolding trajectories were obtained from allatom MD simulations performed by D.E. Shaw Research^{38}. Data were subsampled by a factor of 5 to reduce the size. For some proteins, residues at the flexible tails of proteins were removed from the dataset to reduce noise. Pairwise distances between carbon alpha atoms on the protein backbone were taken, with a cutoff of 6–8 Å, depending on the size of the protein; any distance above the threshold was taken to be equal to the threshold. This vector of pairwise distances was used as input to PCA, to reduce dimension. The first five principle components of the protein data were found to be sufficient for inference of energy landscapes and transition networks (Supplementary Fig. 1).
Generegulatory network simulations
Generegulatory network motifs were simulated using a Gillespie SSA in the SimBiology toolbox in Matlab. A full list of the reactions simulated for each motif, as well as the values of the parameters used, is given in the Supplementary Methods and in the simulation code, which is available from Github (see Code availability).
Reporting summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article.
Data availability
Two publicly available datasets were used in this study. Proteinfolding trajectories^{38} are available from D.E. Shaw Research (https://www.deshawresearch.com/). HIV sequences^{50} are available from https://hiv.biozentrum.unibas.ch/. Generegulatory network simulation data are available upon request, or can be generated by running the simulation code available from Github (see Code availability).
Code availability
The source code used in this study to learn a dynamical transition network and mean first passage times from a Gaussian mixture model is publicly available from Github (https://github.com/philippearce/learningdynamical). Also included are all data processing codes required to convert the raw data used in this study into the appropriate format and the simulation code for the generegulatory network simulations.
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Acknowledgements
We thank D.E. Shaw Research for proteinfolding trajectories and Stefano PianaAgostinetti of D.E. Shaw Research for helpful discussions. This work was supported by the Royal Society International Exchanges award IE160909 (H.K. and J.D.), a summer research studentship from the Biophysical Sciences Institute of Durham University (A.K.), and a Complex Systems Scholar Award from the James S. McDonnell Foundation (J.D.).
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P.P., F.G.W., A.F., H.K. and J.D. developed the theory. P.P., F.G.W., A.K. and H.K. wrote the code for learning a transition network. P.P. wrote the simulation codes and applied the code for learning a transition network to datasets. P.P., F.G.W., A.F., H.K. and J.D. wrote the paper. H.K. and J.D. conceived the study and provided supervision.
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Pearce, P., Woodhouse, F.G., Forrow, A. et al. Learning dynamical information from static protein and sequencing data. Nat Commun 10, 5368 (2019). https://doi.org/10.1038/s4146701913307x
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