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Cells with the same genome can exist in different phenotypes. and can change between distinct phenotypes when subject to specific stimuli and microenvironments. Some examples include cell differentiation during development, reprogramming for induced pluripotent stem cells and transdifferentiation, cancer metastasis and fibrosis development. The regulation and dynamics of cell phenotypic conversion is a fundamental problem in biology, and has a long history of being studied within the formalism of dynamical systems. A main challenge for mechanism-driven modeling studies is acquiring sufficient amount of quantitative information for constraining model parameters. Advances in quantitative approaches, especially high throughput single-cell techniques, have accelerated the emergence of a new direction for reconstructing the governing dynamical equations of a cellular system from quantitative single-cell data, beyond the dominant statistical approaches. Here I review a selected number of recent studies using live- and fixed-cell data and provide my perspective on future development., Comment: 38 pages, 3 figures

Phenotype transition takes place in many biological processes such as differentiation, and understanding how a cell reprograms its global gene expression profile is a problem of rate theories. A cell phenotype transition accompanies with switching of expression rates of clusters of genes, analogous to domain flipping in an Ising system. Here through analyzing single cell RNA sequencing data in the framework of transition path theory, we set to study how such a genome-wide expression program switching proceeds in three different cell transition processes. For each process after reconstructing a Markov transition model in the cell state space, we formed an ensemble of shortest paths connecting the initial and final cell states, reconstructed a reaction coordinate describing the transition progression, and inferred the gene regulation network (GRN) along the reaction coordinate. In all three processes we observed common pattern that the frustration of gene regulatory network (GRN), defined as overall confliction between the regulation received by genes and their expression states, first increases then decreases when approaching a new phenotype. The results support a mechanism of concerted silencing of genes that are active in the initial phenotype and activation of genes that are active in the final phenotype.

Epithelial-Mesenchymal Transition (EMT), and the corresponding reverse process, Mesenchymal-Epithelial Transition (MET), are dynamic and reversible cellular programs orchestrated by many changes at biochemical and morphological levels. A recent surge in identifying the molecular mechanisms underlying EMT/MET has led to the development of various mathematical models that have contributed to our improved understanding of dynamics at single-cell and population levels: a) multi-stability (how many phenotypes can cells attain en route EMT/MET?), b) reversibility/irreversibility (what time and/or concentration of an EMT inducer marks the 'tipping point' when cells induced to undergo EMT cannot revert?), c) symmetry in EMT/MET (do cells take the same path while reverting as they took during the induction of EMT?), and d) non-cell autonomous mechanisms (how does a cell undergoing EMT alter the tendency of its neighbors to undergo EMT?). These dynamical traits may facilitate a heterogeneous response within a cell population undergoing EMT/MET. Here, we present a few examples of designing different mathematical models that can contribute to decoding EMT/MET dynamics.

Over the past 50 years, the use of mathematical models, derived from physical reasoning, to describe molecular and cellular systems has evolved from an art of the few to a cornerstone of biological inquiry. George Oster stood out as a pioneer of this paradigm shift from descriptive to quantitative biology not only through his numerous research accomplishments, but also through the many students and postdocs he mentored over his long career. Those of us fortunate enough to have worked with George agree that his sharp intellect, physical intuition and passion for scientific inquiry not only inspired us as scientists but also greatly influenced the way we conduct research. We would like to share a few important lessons we learned from George in honor of his memory and with the hope that they may inspire future generations of scientists., Comment: 22 pages, 3 figures, accepted in Molecular Biology of the Cell

Many cellular responses to surrounding cues require temporally concerted transcriptional regulation of multiple genes. In prokaryotic cells, a single-input-module motif with one transcription factor regulating multiple target genes can generate coordinated gene expression. In eukaryotic cells, transcriptional activity of a gene is affected by not only transcription factors but also the epigenetic modifications and three-dimensional chromosome structure of the gene. To examine how local gene environment and transcription factor regulation are coupled, we performed a combined analysis of time-course RNA-seq data of TGF-\b{eta} treated MCF10A cells and related epigenomic and Hi-C data. Using Dynamic Regulatory Events Miner (DREM), we clustered differentially expressed genes based on gene expression profiles and associated transcription factors. Genes in each class have similar temporal gene expression patterns and share common transcription factors. Next, we defined a set of linear and radial distribution functions, as used in statistical physics, to measure the distributions of genes within a class both spatially and linearly along the genomic sequence. Remarkably, genes within the same class despite sometimes being separated by tens of million bases (Mb) along genomic sequence show a significantly higher tendency to be spatially close despite sometimes being separated by tens of Mb along the genomic sequence than those belonging to different classes do. Analyses extended to the process of mouse nervous system development arrived at similar conclusions. Future studies will be able to test whether this spatial organization of chromosomes contributes to concerted gene expression., Comment: 30 pages, 9 figures, accepted in PLoS Computational Biology

The transition between epithelial and mesenchymal (EMT) is a fundamental cellular process that plays critical roles in development, cancer metastasis, and tissue wound healing. EMT is not a binary process but involves multiple partial EMT states that give rise to a high degree of cell state plasticity. Here, we first reviewed several studies on theoretical predictions and experimental verification of these intermediate states, the role of partial EMT on kidney fibrosis development, and how quantitative signaling information controls cell commitment to partial or full EMT upon transient signals. Next, we summarized existing knowledge and open questions on the coupling between EMT and other biological processes, such as the cell cycle, epigenetic regulation, stemness, and apoptosis. Taken together, EMT is a model system that has attracted increasing interests for quantitative experimental and theoretical studies., Comment: 37 pages, 6 figures, accepted in Physical Biology

Tissue fibrosis compromises organ function and occurs as a potential long-term outcome in response to acute tissue injuries. Currently, lack of mechanistic understanding prevents effective prevention and treatment of the progression from acute injury to fibrosis. Here, we combined quantitative experimental studies with a mouse kidney injury model and a computational approach to determine how the physiological consequences are determined by the severity of ischemia injury, and to identify how to manipulate Wnt signaling to accelerate repair of ischemic tissue damage while minimizing fibrosis. The study reveals that Wnt-mediated memory of prior injury contributes to fibrosis progression, and ischemic preconditioning reduces the risk of death but increases the risk of fibrosis. Furthermore, we validated the prediction that sequential combination therapy of initial treatment with a Wnt agonist followed by treatment with a Wnt antagonist can reduce both the risk of death and fibrosis in response to acute injuries.

Single cell segmentation is critical and challenging in live cell imaging data analysis. Traditional image processing methods and tools require time-consuming and labor-intensive efforts of manually fine-tuning parameters. Slight variations of image setting may lead to poor segmentation results. Recent development of deep convolutional neural networks(CNN) provides a potentially efficient, general and robust method for segmentation. Most existing CNN-based methods treat segmentation as a pixel-wise classification problem. However, three unique problems of cell images adversely affect segmentation accuracy: lack of established training dataset, few pixels on cell boundaries, and ubiquitous blurry features. The problem becomes especially severe with densely packed cells, where a pixel-wise classification method tends to identify two neighboring cells with blurry shared boundary as one cell, leading to poor cell count accuracy and affecting subsequent analysis. Here we developed a different learning strategy that combines strengths of CNN and watershed algorithm. The method first trains a CNN to learn Euclidean distance transform of binary masks corresponding to the input images. Then another CNN is trained to detect individual cells in the Euclidean distance transform. In the third step, the watershed algorithm takes the outputs from the previous steps as inputs and performs the segmentation. We tested the combined method and various forms of the pixel-wise classification algorithm on segmenting fluorescence and transmitted light images. The new method achieves similar pixel accuracy but significant higher cell count accuracy than pixel-wise classification methods do, and the advantage is most obvious when applying on noisy images of densely packed cells., Comment: 5 figures, 3 supplemental figures

miRNAs serve as crucial post-transcriptional regulators in various essential cell fate decision. However, the contribution of the mRNA-miRNA mutual regulation to bistability is not fully understood. Here, we built a set of mathematical models of mRNA-miRNA interactions and systematically analyzed the sensitivity of response curves under various conditions. First, we found that mRNA-miRNA reciprocal regulation could manifest ultrasensitivity to subserve the generation of bistability when equipped with a positive feedback loop. Second, the region of bistability is expanded by a stronger competing mRNA (ceRNA). Interesting, bistability can be emerged without feedback loop if multiple miRNA binding sites exist on a target mRNA. Thus, we demonstrated the importance of simple mRNA-miRNA reciprocal regulation in cell fate decision., Comment: 32 pages, 5 figures and 7 supporting figures in FEBS Letters, 2016

Multiple-objective optimization is common in biological systems. In the mammalian olfactory system, each sensory neuron stochastically expresses only one out of up to thousands of olfactory receptor (OR) gene alleles; at organism level the types of expressed ORs need to be maximized. Existing models focus only on monoallele activation, and cannot explain recent observations in mutants, especially the reduced global diversity of expressed ORs in G9a/GLP knockouts. In this work we integrated existing information on OR expression, and constructed a comprehensive model that has all its components based on physical interactions. Analyzing the model reveals an evolutionarily optimized three-layer regulation mechanism, which includes zonal segregation, epigenetic barrier crossing coupled to a negative feedback loop that mechanistically differs from previous theoretical proposals, and a previously unidentified enhancer competition step. This model not only recapitulates monoallelic OR expression, but also elucidates how the olfactory system maximizes and maintains the diversity of OR expression, and has multiple predictions validated by existing experimental results. Through making analogy to a physical system with thermally activated barrier crossing and comparative reverse engineering analyses, the study reveals that the olfactory receptor selection system is optimally designed, and particularly underscores cooperativity and synergy as a general design principle for multi-objective optimization in biology., Comment: 10 pages, 6 figures, Proceedings of the National Academy of Sciences 2016

How do mammalian cells that share the same genome exist in notably distinct phenotypes, exhibiting differences in morphology, gene expression patterns, and epigenetic chromatin statuses? Furthermore how do cells of different phenotypes differentiate reproducibly from a single fertilized egg? These are fundamental problems in developmental biology. Epigenetic histone modifications play an important role in the maintenance of different cell phenotypes. The exact molecular mechanism for inheritance of the modification patterns over cell generations remains elusive. The complexity comes partly from the number of molecular species and the broad time scales involved. In recent years mathematical modeling has made significant contributions on elucidating the molecular mechanisms of DNA methylation and histone covalent modification inheritance. We will pedagogically introduce the typical procedure and some technical details of performing a mathematical modeling study, and discuss future developments., Comment: 36 pages, 4 figures, 2 tables, book chapter

Epigenetic histone modifications play an important role in the maintenance of different cell phenotypes. The exact molecular mechanism for inheritance of the modification patterns over cell generations remains elusive. We construct a Potts-type model based on experimentally observed nearest-neighbor enzyme lateral interactions and nucleosome covalent modification state biased enzyme recruitment. The model can lead to effective nonlocal interactions among nucleosomes suggested in previous theoretical studies, and epigenetic memory is robustly inheritable against stochastic cellular processes., Comment: Published in PRL

Epithelial to mesenchymal transition (EMT) plays important roles in embryonic development, tissue regeneration and cancer metastasis. While several feedback loops have been shown to regulate EMT, it remains elusive how they coordinately modulate EMT response to TGF-\beta treatment. We construct a mathematical model for the core regulatory network controlling TGF-\beta-induced EMT. Through deterministic analyses and stochastic simulations, we show that EMT is a sequential two-step program that an epithelial cell first transits to partial EMT then to the mesenchymal state, depending on the strength and duration of TGF-\beta stimulation. Mechanistically the system is governed by coupled reversible and irreversible bistable switches. The SNAIL1/miR-34 double negative feedback loop is responsible for the reversible switch and regulates the initiation of EMT, while the ZEB/miR-200 feedback loop is accountable for the irreversible switch and controls the establishment of the mesenchymal state. Furthermore, an autocrine TGF-\beta/miR-200 feedback loop makes the second switch irreversible, modulating the maintenance of EMT. Such coupled bistable switches are robust to parameter variation and molecular noise. We provide a mechanistic explanation on multiple experimental observations. The model makes several explicit predictions on hysteretic dynamic behaviors, system response to pulsed stimulation and various perturbations, which can be straightforwardly tested., Comment: 32 pages, 8 figures, accepted by Biophysical Journal

The Zwanzig-Mori pro jection formalism is widely used in studying systems with many degrees of freedom. Recently Xing and Kim used the pro jection formalism and derived the generalized Langevin equations (GLEs) for a general stochastic system not necessarily obeying detailed balance. In this study we develop a numerical procedure to reconstruct the GLEs from data. Numerical tests on two biological networks show remarkable agreement between the results calculated from the reconstructed GLEs and those of full model simulations. We suggest that the procedure can be applied in model reduction and a novel way of nonlinear time series analysis., Comment: 9 pages, 4 figures

How do mammalian cells that share the same genome exist in notably distinct phenotypes, exhibiting differences in morphology, gene expression patterns, and epigenetic chromatin statuses? Furthermore how do cells of different phenotypes differentiate reproducibly from a single fertilized egg? These fundamental questions are closely related to a deeply rooted paradigm in developmental biology that cell differentiation is irreversible. Yet, recently a growing body of research suggests the possibility of cell reprogramming, which offers the potential for us to convert one type of cell into another. Despite the significance of quantitative understandings of cell reprogramming, theoretical efforts often suffer from the complexity of large circuits maintaining cell phenotypes coupled at many different epigenetic and gene regulation levels. To capture the global architecture of cell phenotypes, we propose an "epigenetic state network" approach that translates the classical concept of an epigenetic landscape into a simple-yet-predictive mathematical model. As a testing case, we apply the approach to the reprogramming of fibroblasts (FB) to cardiomyocytes (CM). The epigenetic state network for this case predicts three major pathways of reprogramming. One pathway goes by way of induced pluripotent stem cells (iPSC) and continues on to the normal pathway of cardiomyocyte differentiation. The other two pathways involve transdifferentiation (TD) either indirectly through cardiac progenitor (CP) cells or directly from fibroblast to cardiomyocyte. Numerous experimental observations support the predicted states and pathways., Comment: 18 pages,4 figures, 2 tables

The innate immune system, acting as the first line of host defense, senses and adapts to foreign challenges through complex intracellular and intercellular signaling networks. Endotoxin tolerance and priming elicited by macrophages are classic examples of the complex adaptation of innate immune cells. Upon repetitive exposures to different doses of bacterial endotoxin (lipopolysaccharide) or other stimulants, macrophages show either suppressed or augmented inflammatory responses compared to a single exposure to the stimulant. Endotoxin tolerance and priming are critically involved in both immune homeostasis and the pathogenesis of diverse inflammatory diseases. However, the underlying molecular mechanisms are not well understood. By means of a computational search through the parameter space of a coarse-grained three-node network with a two-stage Metropolis sampling approach, we enumerated all the network topologies that can generate priming or tolerance. We discovered three major mechanisms for priming (pathway synergy, suppressor deactivation, activator induction) and one for tolerance (inhibitor persistence). These results not only explain existing experimental observations, but also reveal intriguing test scenarios for future experimental studies to clarify mechanisms of endotoxin priming and tolerance., Comment: 15 pages, 8 figures, submitted

The bacterial flagellar motor plays a crucial role in both bacterial locomotion and chemotaxis. Recent experiments reveal that the switching dynamics of the motor depends on the motor rotation speed, and thus the motor torque, non-monotonically. Here we present a unified mathematical model which models motor torque generation based on experimental torque-speed curves and torque-dependent switching based on the conformational spread model. The model successfully reproduces the observed switching rate as a function of the rotation speed, and provides a generic physical explanation independent of most details. A stator affects the switching dynamics through two mechanisms: accelerating the conformation flipping rates of individual rotor switching units, which favours slower motor speed and thus increasing torque; and affecting more switching units within unit time, which favours faster speed. Consequently, the switching rate shows a maximum at intermediate speed. Our model predicts that a motor switches more often with more stators. The load-switching relation may serve as a mechanism for sensing the physical environment, similar to the chemotaxis system for sensing the chemical environment. It may also coordinate the switch dynamics of motors within a cell., Comment: 23 pages, 5 figures

A bacterial colony may develop a small number of cells genetically identical to, but phenotypically different from other normally growing bacteria. These so-called persister cells keep themselves in a dormant state and thus are insensitive to antibiotic treatment, resulting in serious problems of drug resistance. In this paper, we proposed a novel strategy to "kill" persister cells by triggering them to switch, in a fast and synchronized way, into normally growing cells that are susceptible to antibiotics. The strategy is based on resonant activation (RA), a well-studied phenomenon in physics where the internal noise of a system can constructively facilitate fast and synchronized barrier crossings. Through stochastic Gilliespie simulation with a generic toggle switch model, we demonstrated that RA exists in the phenotypic switching of a single bacterium. Further, by coupling single cell level and population level simulations, we showed that with RA, one can greatly reduce the time and total amount of antibiotics needed to sterilize a bacterial population. We suggest that resonant activation is a general phenomenon in phenotypic transition, and can find other applications such as cancer therapy., Comment: 21 pages, 12 figures, submitted

The microtubule assembly process has been extensively studied, but the underlying molecular mechanism remains poorly understood. The structure of an artificially generated sheet polymer that alternates two types of lateral contacts and that directly converts into microtubules, has been proposed to correspond to the intermediate sheet structure observed during microtubule assembly. We have studied the self-assembly process of GMPCPP tubulins into sheet and microtubule structures using thermodynamic analysis and stochastic simulations. With the novel assumptions that tubulins can laterally interact in two different forms, and allosterically affect neighboring lateral interactions, we can explain existing experimental observations. At low temperature, the allosteric effect results in the observed sheet structure with alternating lateral interactions as the thermodynamically most stable form. At normal microtubule assembly temperature, our work indicates that a class of sheet structures resembling those observed at low temperature is transiently trapped as an intermediate during the assembly process. This work may shed light on the tubulin molecular interactions, and the role of sheet formation during microtubule assembly., Comment: 43 pages, 13 figures. Submitted; PLoS ONE 2009

A protein undergoes conformational dynamics with multiple time scales, which results in fluctuating enzyme activities. Recent studies in single molecule enzymology have observe this "age-old" dynamic disorder phenomenon directly. However, the single molecule technique has its limitation. To be able to observe this molecular effect with real biochemical functions {\it in situ}, we propose to couple the fluctuations in enzymatic activity to noise propagations in small protein interaction networks such as zeroth order ultra-sensitive phosphorylation-dephosphorylation cycle. We showed that enzyme fluctuations could indeed be amplified by orders of magnitude into fluctuations in the level of substrate phosphorylation | a quantity widely interested in cellular biology. Enzyme conformational fluctuations sufficiently slower than the catalytic reaction turn over rate result in a bimodal concentration distribution of the phosphorylated substrate. In return, this network amplified single enzyme fluctuation can be used as a novel biochemical "reporter" for measuring single enzyme conformational fluctuation rates., Comment: 9 pages, 5 figures

Theoretical studies of nonequilibrium systems are complicated by the lack of a general framework. In this work we first show that a transformation introduced by Ao recently (J. Phys. A {\bf 37}, L25 (2004)) is related to previous works of Graham (Z. Physik B {\bf 26}, 397 (1977)) and Eyink {\it et al.} (J. Stat. Phys. {\bf 83}, 385 (1996)), which can also be viewed as the generalized application of the Helmholtz theorem in vector calculus. We then show that systems described by ordinary stochastic differential equations with white noise can be mapped to thermostated Hamiltonian systems. A steady-state of a dissipative system corresponds to the equilibrium state of the corresponding Hamiltonian system. These results provides a solid theoretical ground for corresponding studies on nonequilibrium dynamics, especially on nonequilibrium steady state. The mapping permits the application of established techniques and results for Hamiltonian systems to dissipative non-Hamiltonian systems, those for thermodynamic equilibrium states to nonequilibrium steady states. We discuss several implications of the present work., Comment: 18 pages, no figure. final version for publication on J. Phys. A: Math & Theor

Reconstruction of equations of motion from incomplete or noisy data and dimension reduction are two fundamental problems in the study of dynamical systems with many degrees of freedom. For the latter extensive efforts have been made but with limited success to generalize the Zwanzig-Mori projection formalism, originally developed for Hamiltonian systems close to thermodynamic equilibrium, to general non-Hamiltonian systems lacking detailed-balance. One difficulty introduced by such systems is the lack of an invariant measure, needed to define a statistical distribution. Based on a recent discovery that a non-Hamiltonian system defined by a set of stochastic differential equations can be mapped to a Hamiltonian system, we develop such general projection formalism. In the resulting generalized Langevin equations, a set of generalized fluctuation-dissipation relations connect the memory kernel and the random noise terms, analogous to Hamiltonian systems obeying detailed balance. Lacking of these relations restricts previous application of the generalized Langevin formalism. Result of this work may serve as the theoretical basis for further technical developments on model reconstruction with reduced degrees of freedom. We first use an analytically solvable example to illustrate the formalism and the fluctuation-dissipation relation. Our numerical test on a chemical network with end-product inhibition further demonstrates the validity of the formalism. We suggest that the formalism can find wide applications in scientific modeling. Specifically, we discuss potential applications to biological networks. In particular, the method provides a suitable framework for gaining insights into network properties such as robustness and parameter transferability., Comment: 35 pages, 2 figures

The fluctuation-dissipation (F-D) theorem is a fundamental result for systems near thermodynamic equilibrium, and justifies studies between microscopic and macroscopic properties. It states that the nonequilibrium relaxation dynamics is related to the spontaneous fluctuation at equilibrium. Most processes in Nature are out of equilibrium, for which we have limited theory. Common wisdom believes the F-D theorem is violated in general for systems far from equilibrium. Recently we show that dynamics of a dissipative system described by stochastic differential equations can be mapped to that of a thermostated Hamiltonian system, with a nonequilibrium steady state of the former corresponding to the equilibrium state of the latter. Her we derived the corresponding F-D theorem, and tested with several examples. We suggest further studies exploiting the analogy between a general dissipative system appearing in various science branches and a Hamiltonian system. Especially we discussed the implications of this work on biological network studies., Comment: 12 pages, 4 figures, major revision over the first version

The Mori-Zwanzig projection formalism is widely used in studying systems with many degrees of freedom. We used a system-bath Hamiltonian system to show that the Mori's and Zwanzig's projection procedures are mutual limiting cases of each other depending on the size of the projected Hilbert space. We also derived the dynamic equations of collective coordinates of a Hamiltonian system., Comment: 3 pages

In their classical work (Proc. Natl. Acad. Sci. USA, 1981, 78:6840-6844), Goldbeter and Koshland mathematically analyzed a reversible covalent modification system which is highly sensitive to the concentration of effectors. Its signal-response curve appears sigmoidal, constituting a biochemical switch. However, the switch behavior only emerges in the "zero-order region", i.e. when the signal molecule concentration is much lower than that of the substrate it modifies. In this work we showed that the switching behavior can also occur under comparable concentrations of signals and substrates, provided that the signal molecules catalyze the modification reaction in cooperation. We also studied the effect of dynamic disorders on the proposed biochemical switch, in which the enzymatic reaction rates, instead of constant, appear as stochastic functions of time. We showed that the system is robust to dynamic disorder at bulk concentration. But if the dynamic disorder is quasi-static, large fluctuations of the switch response behavior may be observed at low concentrations. Such fluctuation is relevant to many biological functions. It can be reduced by either increasing the conformation interconversion rate of the protein, or correlating the enzymatic reaction rates in the network., Comment: 23 pages, 4 figures, accepted by PLOS ONE

Allosteric regulation is often viewed as thermodynamic in nature. However protein internal motions during an enzymatic reaction cycle can be slow hopping processes over numerous potential barriers. We propose that regulating molecules may function by modifying the nonequilibrium protein dynamics. The theory predicts that an enzyme under the new mechanism has different temperature dependence, waiting time distribution of the turnover cycle, and dynamic fluctuation patterns with and without effector. Experimental tests of the theory are proposed., Comment: accepted by Phys. Rev. Lett. Major revisions were made to fit the style. 4 pages, 2 figures