Protein folding dynamics via quantification of kinematic energy landscape

Multiscale Modeling of Cellular Epigenetic States: Stochasticity in Molecular Networks, Chromatin Folding in Cell Nuclei, and Tissue Pattern Formation of Cells

Genome sequences provide the overall genetic blueprint of cells, but cells possessing the same genome can exhibit diverse phenotypes. There is a multitude of mechanisms controlling cellular epigenetic states and that dictate the behavior of cells. Among these, networks of interacting molecules, often under stochastic control, depending on the specific wirings of molecular components and the physiological conditions, can have a different landscape of cellular states. In addition, chromosome folding in three-dimensional space provides another important control mechanism for selective activation and repression of gene expression. Fully differentiated cells with different properties grow, divide, and interact through mechanical forces and communicate through signal transduction, resulting in the formation of complex tissue patterns. Developing quantitative models to study these multi-scale phenomena and to identify opportunities for improving human health requires development of theoretical models, algorithms, and computational tools. Here we review recent progress made in these important directions.

Kinetic mechanism of conformational switch between bistable RNA hairpins

Transitions between the different conformational states play a critical role in many RNA catalytic and regulatory functions. In this study, we use the Kinetic Monte Carlo method to investigate the kinetic mechanism for the conformational switches between bistable RNA hairpins. We find three types of conformational switch pathways for RNA hairpins: refolding after complete unfolding, folding through basepair-exchange pathways and through pseudoknot-assisted pathways, respectively. The result of the competition between the three types of pathways depends mainly on the location of the rate-limiting base stacks (such as the GC base stacks) in the structures. Depending on the structural relationships between the two bistable hairpins, the conformational switch can follow single or multiple dominant pathways. The predicted folding pathways are supported by the activation energy results derived from the Arrhenius plot as well as the NMR spectroscopy data.

Constrained proper sampling of conformations of transition state ensemble of protein folding

Characterizing the conformations of protein in the transition state ensemble (TSE) is important for studying protein folding. A promising approach pioneered by Vendruscolo et al. [Nature (London) 409, 641 (2001)] to study TSE is to generate conformations that satisfy all constraints imposed by the experimentally measured φ values that provide information about the native likeness of the transition states. Faísca et al. [J. Chem. Phys. 129, 095108 (2008)] generated conformations of TSE based on the criterion that, starting from a TS conformation, the probabilities of folding and unfolding are about equal through Markov Chain Monte Carlo (MCMC) simulations. In this study, we use the technique of constrained sequential Monte Carlo method [Lin et al., J. Chem. Phys. 129, 094101 (2008); Zhang et al. Proteins 66, 61 (2007)] to generate TSE conformations of acylphosphatase of 98 residues that satisfy the φ-value constraints, as well as the criterion that each conformation has a folding probability of 0.5 by Monte Carlo simulations. We adopt a two stage process and first generate 5000 contact maps satisfying the φ-value constraints. Each contact map is then used to generate 1000 properly weighted conformations. After clustering similar conformations, we obtain a set of properly weighted samples of 4185 candidate clusters. Representative conformation of each of these cluster is then selected and 50 runs of Markov chain Monte Carlo (MCMC) simulation are carried using a regrowth move set. We then select a subset of 1501 conformations that have equal probabilities to fold and to unfold as the set of TSE. These 1501 samples characterize well the distribution of transition state ensemble conformations of acylphosphatase. Compared with previous studies, our approach can access much wider conformational space and can objectively generate conformations that satisfy the φ-value constraints and the criterion of 0.5 folding probability without bias. In contrast to previous studies, our results show that transition state conformations are very diverse and are far from nativelike when measured in cartesian root-mean-square deviation (cRMSD): the average cRMSD between TSE conformations and the native structure is 9.4 Å for this short protein, instead of 6 Å reported in previous studies. In addition, we found that the average fraction of native contacts in the TSE is 0.37, with enrichment in native-like β-sheets and a shortage of long range contacts, suggesting such contacts form at a later stage of folding. We further calculate the first passage time of folding of TSE conformations through calculation of physical time associated with the regrowth moves in MCMC simulation through mapping such moves to a Markovian state model, whose transition time was obtained by Langevin dynamics simulations. Our results indicate that despite the large structural diversity of the TSE, they are characterized by similar folding time. Our approach is general and can be used to study TSE in other macromolecules.

Probability landscape of heritable and robust epigenetic state of lysogeny in phage lambda

Computational studies of biological networks can help to identify components and wirings responsible for observed phenotypes. However, studying stochastic networks controlling many biological processes is challenging. Similar to Schrödinger's equation in quantum mechanics, the chemical master equation (CME) provides a basic framework for understanding stochastic networks. However, except for simple problems, the CME cannot be solved analytically. Here we use a method called discrete chemical master equation (dCME) to compute directly the full steady-state probability landscape of the lysogeny maintenance network in phage lambda from its CME. Results show that wild-type phage lambda can maintain a constant level of repressor over a wide range of repressor degradation rate and is stable against UV irradiation, ensuring heritability of the lysogenic state. Furthermore, it can switch efficiently to the lytic state once repressor degradation increases past a high threshold by a small amount. We find that beyond bistability and nonlinear dimerization, cooperativity between repressors bound to O(R)1 and O(R)2 is required for stable and heritable epigenetic state of lysogeny that can switch efficiently. Mutants of phage lambda lack stability and do not possess a high threshold. Instead, they are leaky and respond to gradual changes in degradation rate. Our computation faithfully reproduces the hair triggers for UV-induced lysis observed in mutants and the limitation in robustness against mutations. The landscape approach computed from dCME is general and can be applied to study broad issues in systems biology.

Computational Cellular Dynamics Based on the Chemical Master Equation: A Challenge for Understanding Complexity

Modern molecular biology has always been a great source of inspiration for computational science. Half a century ago, the challenge from understanding macromolecular dynamics has led the way for computations to be part of the tool set to study molecular biology. Twenty-five years ago, the demand from genome science has inspired an entire generation of computer scientists with an interest in discrete mathematics to join the field that is now called bioinformatics. In this paper, we shall lay out a new mathematical theory for dynamics of biochemical reaction systems in a small volume (i.e., mesoscopic) in terms of a stochastic, discrete-state continuous-time formulation, called the chemical master equation (CME). Similar to the wavefunction in quantum mechanics, the dynamically changing probability landscape associated with the state space provides a fundamental characterization of the biochemical reaction system. The stochastic trajectories of the dynamics are best known through the simulations using the Gillespie algorithm. In contrast to the Metropolis algorithm, this Monte Carlo sampling technique does not follow a process with detailed balance. We shall show several examples how CMEs are used to model cellular biochemical systems. We shall also illustrate the computational challenges involved: multiscale phenomena, the interplay between stochasticity and nonlinearity, and how macroscopic determinism arises from mesoscopic dynamics. We point out recent advances in computing solutions to the CME, including exact solution of the steady state landscape and stochastic differential equations that offer alternatives to the Gilespie algorithm. We argue that the CME is an ideal system from which one can learn to understand "complex behavior" and complexity theory, and from which important biological insight can be gained.

Perturbation-based Markovian transmission model for probing allosteric dynamics of large macromolecular assembling: a study of GroEL-GroES

Large macromolecular assemblies are often important for biological processes in cells. Allosteric communications between different parts of these molecular machines play critical roles in cellular signaling. Although studies of the topology and fluctuation dynamics of coarse-grained residue networks can yield important insights, they do not provide characterization of the time-dependent dynamic behavior of these macromolecular assemblies. Here we develop a novel approach called Perturbation-based Markovian Transmission (PMT) model to study globally the dynamic responses of the macromolecular assemblies. By monitoring simultaneous responses of all residues (>8,000) across many (>6) decades of time spanning from the initial perturbation until reaching equilibrium using a Krylov subspace projection method, we show that this approach can yield rich information. With criteria based on quantitative measurements of relaxation half-time, flow amplitude change, and oscillation dynamics, this approach can identify pivot residues that are important for macromolecular movement, messenger residues that are key to signal mediating, and anchor residues important for binding interactions. Based on a detailed analysis of the GroEL-GroES chaperone system, we found that our predictions have an accuracy of 71-84% judged by independent experimental studies reported in the literature. This approach is general and can be applied to other large macromolecular machineries such as the virus capsid and ribosomal complex.

Statistical geometry of lattice chain polymers with voids of defined shapes: sampling with strong constraints

Proteins contain many voids, which are unfilled spaces enclosed in the interior. A few of them have shapes compatible to ligands and substrates and are important for protein functions. An important general question is how the need for maintaining functional voids is influenced by, and affects other aspects of proteins structures and properties (e.g., protein folding stability, kinetic accessibility, and evolution selection pressure). In this paper, we examine in detail the effects of maintaining voids of different shapes and sizes using two-dimensional lattice models. We study the propensity for conformations to form a void of specific shape, which is related to the entropic cost of void maintenance. We also study the location that voids of a specific shape and size tend to form, and the influence of compactness on the formation of such voids. As enumeration is infeasible for long chain polymer, a key development in this work is the design of a novel sequential Monte Carlo strategy for generating large number of sample conformations under very constraining restrictions. Our method is validated by comparing results obtained from sampling and from enumeration for short polymer chains. We succeeded in accurate estimation of entropic cost of void maintenance, with and without an increasing number of restrictive conditions, such as loops forming the wall of void with fixed length, with additionally fixed starting position in the sequence. Additionally, we have identified the key structural properties of voids that are important in determining the entropic cost of void formation. We have further developed a parametric model to predict quantitatively void entropy. Our model is highly effective, and these results indicate that voids representing functional sites can be used as an improved model for studying the evolution of protein functions and how protein function relates to protein stability.

Perturbation-based Markovian Transmission Model for macromolecular machinery in cell

The study of the dynamics of a complex system is an important problem that includes large macromolecular complexes, molecular interaction networks, and cell functional modules. Large macromolecular complexes in cellular machinery can be modeled as a connected network, as in the elastic or Gaussian network models as demonstrated by Bahar and colleagues. Here we propose the Perturbation-based Markovian Transmission Model for studying the dynamics of signal transmission in macromolecular machinery. The initial perturbation is transmitted by a Markovian processes, and the dynamics of the probability flow is analytically solved using the master equation. Due to the large size of macromolecular complexes, it is very difficult to obtain analytical time-dependent Markovian dynamics of all atoms from the first perturbation until stationary state. To overcome it, we decrease the level of complexity of the transition matrix using a Krylov subspace method. This method is equivalent to integrating all eigen modes, and we show it can provide a globally accurate solution to the dynamics problem of signal transmission for very large macromolecular complexes with reasonable computational time. We give results of the dynamics of the GroEL-GroES chaperone system by applying uniform perturbation to all residues. We are able to identify experimentally found important residues and provide a set of predicted pivot, messenger, and effector residues, each with distinct dynamic behavior. Further results of selective perturbation on the surface of ATP binding pocket identifies the path of maximal probability flow of signal.

Predicting protein folding rates from geometric contact and amino acid sequence

Protein folding speeds are known to vary over more than eight orders of magnitude. Plaxco, Simons, and Baker (see References) first showed a correlation of folding speed with the topology of the native protein. That and subsequent studies showed, if the native structure of a protein is known, its folding speed can be predicted reasonably well through a correlation with the "localness" of the contacts in the protein. In the present work, we develop a related measure, the geometric contact number, N (alpha), which is the number of nonlocal contacts that are well-packed, by a Voronoi criterion. We find, first, that in 80 proteins, the largest such database of proteins yet studied, N (alpha) is a consistently excellent predictor of folding speeds of both two-state fast folders and more complex multistate folders. Second, we show that folding rates can also be predicted from amino acid sequences directly, without the need to know the native topology or other structural properties.

Optimal enumeration of state space of finitely buffered stochastic molecular networks and exact computation of steady state landscape probability

BACKGROUND

Stochasticity plays important roles in many molecular networks when molecular concentrations are in the range of 0.1 muM to 10nM (about 100 to 10 copies in a cell). The chemical master equation provides a fundamental framework for studying these networks, and the time-varying landscape probability distribution over the full microstates, i.e., the combination of copy numbers of molecular species, provide a full characterization of the network dynamics. A complete characterization of the space of the microstates is a prerequisite for obtaining the full landscape probability distribution of a network. However, there are neither closed-form solutions nor algorithms fully describing all microstates for a given molecular network.

RESULTS

We have developed an algorithm that can exhaustively enumerate the microstates of a molecular network of small copy numbers under the condition that the net gain in newly synthesized molecules is smaller than a predefined limit. We also describe a simple method for computing the exact mean or steady state landscape probability distribution over microstates. We show how the full landscape probability for the gene networks of the self-regulating gene and the toggle-switch in the steady state can be fully characterized. We also give an example using the MAPK cascade network. Data and server will be available at URL: http://scsb.sjtu.edu.cn/statespace.

CONCLUSION

Our algorithm works for networks of small copy numbers buffered with a finite copy number of net molecules that can be synthesized, regardless of the reaction stoichiometry, and is optimal in both storage and time complexity. The algorithm can also be used to calculate the rates of all transitions between microstates from given reactions and reaction rates. The buffer size is limited by the available memory or disk storage. Our algorithm is applicable to a class of biological networks when the copy numbers of molecules are small and the network is closed, or the network is open but the net gain in newly synthesized molecules does not exceed a predefined buffer capacity. For these networks, our method allows full stochastic characterization of the mean landscape probability distribution, and the steady state when it exists.

Loop-closure principles in protein folding

Simple theoretical concepts and models have been helpful to understand the folding rates and routes of single-domain proteins. As reviewed in this article, a physical principle that appears to underly these models is loop closure.

Biopolymer structure simulation and optimization via fragment regrowth Monte Carlo

An efficient exploration of the configuration space of a biopolymer is essential for its structure modeling and prediction. In this study, the authors propose a new Monte Carlo method, fragment regrowth via energy-guided sequential sampling (FRESS), which incorporates the idea of multigrid Monte Carlo into the framework of configurational-bias Monte Carlo and is suitable for chain polymer simulations. As a by-product, the authors also found a novel extension of the Metropolis Monte Carlo framework applicable to all Monte Carlo computations. They tested FRESS on hydrophobic-hydrophilic (HP) protein folding models in both two and three dimensions. For the benchmark sequences, FRESS not only found all the minimum energies obtained by previous studies with substantially less computation time but also found new lower energies for all the three-dimensional HP models with sequence length longer than 80 residues.

An optimal algorithm for enumerating state space of stochastic molecular networks with small copy numbers of molecules

Stochasticity plays central role in molecular networks of small copy numbers, including those important in protein synthesis and gene regulation. The combination of copy numbers of molecular species defines the microscopic state of molecular interactions. With this formulation, nonlinear reactions can be effectively modeled through chemical master equations. However, currently little is known about the state space associated with stochastic networks, other than the defeatist admission that it is exponentially large. There is neither closed-form solution nor computational algorithm that can effectively characterize the state space of molecular networks. Such a characterization is a prerequisite for directly solving the chemical master equation. In this study, we describe an algorithm that can exhaustively characterize all possible states of a molecular networks with small copy numbers of species for a given initial condition. Our algorithm works for networks of arbitrary stoichiometry, and is optimal in both storage and time complexity. It allows the approach of solving chemical master equation to be applicable to a larger class of stochastic molecular networks. We show an example of application of our method to the MAPK cascade network.