Discrete-state Stochastic Model Research Articles

Molecular mechanisms of precise timing in cell lysis

Many biological systems exhibit precise timing of events, and one of the most known examples is cell lysis, which is a process of breaking bacterial host cells in the virus infection cycle. However, the underlying microscopic picture of precise timing remains not well understood. We present a novel theoretical approach to explain the molecular mechanisms of effectively deterministic dynamics in biological systems. Our hypothesis is based on the idea of stochastic coupling between relevant underlying biophysical and biochemical processes that lead to noise cancellation. To test this hypothesis, we introduced a minimal discrete-state stochastic model to investigate how holin proteins produced by bacteriophages break the inner membranes of gram-negative bacteria. By explicitly solving this model, the dynamic properties of cell lysis are fully evaluated, and theoretical predictions quantitatively agree with available experimental data for both wild-type and holin mutants. It is found that the observed threshold-like behavior is a result of the balance between holin proteins entering the membrane and leaving the membrane during the lysis. Theoretical analysis suggests that the cell lysis achieves precise timing for wild-type species by maximizing the number of holins in the membrane and narrowing their spatial distribution. In contrast, for mutated species, these conditions are not satisfied. Our theoretical approach presents a possible molecular picture of precise dynamic regulation in intrinsically random biological processes.

Role of Nucleosome Sliding in the Protein Target Search for Covered DNA Sites.

Associations of transcription factors (TFs) with specific sites on DNA initiate major cellular processes. But DNA in eukaryotic cells is covered by nucleosomes which prevent TFs from binding. However, nucleosome structures on DNA are not static and exhibit breathing and sliding. We develop a theoretical framework to investigate the effect of nucleosome sliding on a protein target search. By analysis of a discrete-state stochastic model of nucleosome sliding, search dynamics are explicitly evaluated. It is found that for long sliding lengths the target search dynamics are faster for normal TFs that cannot enter the nucleosomal DNA. But for more realistic short sliding lengths, the so-called pioneer TFs, which can invade nucleosomal DNA, locate specific sites faster. It is also suggested that nucleosome breathing, which is a faster process, has a stronger effect on protein search dynamics than that of nucleosome sliding. Theoretical arguments to explain these observations are presented.

Microscopic Mechanism of Macromolecular Crowder-Assisted DNA Capture and Translocation through Biological Nanopores.

Biological nanopore sensors are widely used for genetic sequencing as nucleic acids and other molecules translocate through them across membranes. Recent studies have shown that the transport of these polymers through nanopores is strongly influenced by macromolecular bulk crowders. By using poly(ethylene glycol) (PEG) molecules as crowders, experiments have shown an increase in the capture rates and translocation times of polymers through an α-hemolysin (αHL) nanopore, which provides high-throughput signals and accurate sensing. A clear molecular-level understanding of how the presence of PEGs offers such desirable outcomes in nanopore sensing is still missing. In this work, we present a new theoretical approach to probe the effect of PEG crowders on DNA capture and translocation through the αHL nanopore. We develop an exactly solvable discrete-state stochastic model based on the cooperative partitioning of individual polycationic PEGs within the cavity of the αHL nanopore. It is argued that the apparent electrostatic interactions between the DNA and PEGs control all of the dynamic processes. Our analytical predictions find excellent agreements with existing experiments, thereby strongly supporting our theory.

The role of spatial structures of tissues in cancer initiation dynamics

It is widely believed that biological tissues evolved to lower the risks of cancer development. One of the specific ways to minimize the chances of tumor formation comes from proper spatial organization of tissues. However, the microscopic mechanisms of underlying processes remain not fully understood. We present a theoretical investigation on the role of spatial structures in cancer initiation dynamics. In our approach, the dynamics of single mutation fixations are analyzed using analytical calculations and computer simulations by mapping them to Moran processes on graphs with different connectivity that mimic various spatial structures. It is found that while the fixation probability is not affected by modifying the spatial structures of the tissues, the fixation times can change dramatically. The slowest dynamics is observed in ‘quasi-one-dimensional’ structures, while the fastest dynamics is observed in ‘quasi-three-dimensional’ structures. Theoretical calculations also suggest that there is a critical value of the degree of graph connectivity, which mimics the spatial dimension of the tissue structure, above which the spatial structure of the tissue has no effect on the mutation fixation dynamics. An effective discrete-state stochastic model of cancer initiation is utilized to explain our theoretical results and predictions. Our theoretical analysis clarifies some important aspects on the role of the tissue spatial structures in the cancer initiation processes.

How Pioneer Transcription Factors Search for Target Sites on Nucleosomal DNA.

All major biological processes start after protein molecules known as transcription factors detect specific regulatory sequences on DNA and initiate genetic expression by associating to them. But in eukaryotic cells, much of the DNA is covered by nucleosomes and other chromatin structures, preventing transcription factors from binding to their targets. At the same time, experimental studies show that there are several classes of proteins, called "pioneer transcription factors", that are able to reach the targets on nucleosomal DNA; however, the underlying microscopic mechanisms remain not well understood. We propose a new theoretical approach that might explain how pioneer transcription factors can find their targets. It is argued that pioneer transcription factors might weaken the interactions between the DNA and nucleosome by substituting them with similar interactions between transcription factors and DNA. Using this idea, we develop a discrete-state stochastic model that allows for exact calculations of target search dynamics on nucleosomal DNA using first-passage probabilities approach. It is found that the target search on nuclesomal DNA for pioneer transcription factors might be significantly accelerated while the search is slower on naked DNA in comparison with normal transcription factors. Our theoretical predictions are supported by Monte Carlo computer simulations, and they also agree with available experimental observations.

Microscopic mechanisms of cooperative communications within single nanocatalysts

Catalysis is a method of accelerating chemical reactions that is critically important for fundamental research as well as for industrial applications. It has been recently discovered that catalytic reactions on metal nanoparticles exhibit cooperative effects. The mechanism of these observations, however, remains not well understood. In this work, we present a theoretical investigation on possible microscopic origin of cooperative communications in nanocatalysts. In our approach, the main role is played by positively charged holes on metal surfaces. A corresponding discrete-state stochastic model for the dynamics of holes is developed and explicitly solved. It is shown that the observed spatial correlation lengths are given by the average distances migrated by the holes before they disappear, while the temporal memory is determined by their lifetimes. Our theoretical approach is able to explain the universality of cooperative communications as well as the effect of external electric fields. Theoretical predictions are in agreement with experimental observations. The proposed theoretical framework quantitatively clarifies some important aspects of the microscopic mechanisms of heterogeneous catalysis.

Kinetic origin of nucleosome invasion by pioneer transcription factors

Recently, a cryo-electron microscopy study has captured different stages of nucleosome breathing dynamics that show partial unwrapping of DNA from histone core to permit transient access to the DNA sites by transcription factors. In practice, however, only a subset of transcription factors named pioneer factors can invade nucleosomes and bind to specific DNA sites to trigger essential DNA metabolic processes. We propose a discrete-state stochastic model that considers the interplay of nucleosome breathing and protein dynamics explicitly and estimate the mean time to search the target DNA sites. It is found that the molecular principle governing the search process on nucleosome is very different compared to that on naked DNA. The pioneer factors minimize their search times on nucleosomal DNA by compensating their nucleosome association rates by dissociation rates. A fine balance between the two presents a tradeoff between their nuclear mobility and error associated with the search process.

Integrated Information in the Spiking-Bursting Stochastic Model.

Integrated information has been recently suggested as a possible measure to identify a necessary condition for a system to display conscious features. Recently, we have shown that astrocytes contribute to the generation of integrated information through the complex behavior of neuron–astrocyte networks. Still, it remained unclear which underlying mechanisms governing the complex behavior of a neuron–astrocyte network are essential to generating positive integrated information. This study presents an analytic consideration of this question based on exact and asymptotic expressions for integrated information in terms of exactly known probability distributions for a reduced mathematical model (discrete-time, discrete-state stochastic model) reflecting the main features of the “spiking–bursting” dynamics of a neuron–astrocyte network. The analysis was performed in terms of the empirical “whole minus sum” version of integrated information in comparison to the “decoder based” version. The “whole minus sum” information may change sign, and an interpretation of this transition in terms of “net synergy” is available in the literature. This motivated our particular interest in the sign of the “whole minus sum” information in our analytical considerations. The behaviors of the “whole minus sum” and “decoder based” information measures are found to bear a lot of similarity—they have mutual asymptotic convergence as time-uncorrelated activity increases, and the sign transition of the “whole minus sum” information is associated with a rapid growth in the “decoder based” information. The study aims at creating a theoretical framework for using the spiking–bursting model as an analytically tractable reference point for applying integrated information concepts to systems exhibiting similar bursting behavior. The model can also be of interest as a new discrete-state test bench for different formulations of integrated information.

Discrete-state Stochastic Modeling of T-cell Activation

One of the most important functions of immune T-cells is to recognize the presence of the pathogen-derived ligands and quickly respond to them while at the same time not to respond to its own ligands. This is known as absolute discrimination, and it is one of the most challenging phenomena to explain in the immune system response. The effectiveness of pathogen detection by T cell receptor is limited by the molecular similarity of foreign and self-peptides since the healthy body produces very similar proteins. So, it is not clear what is the discrimination criteria. We developed a discrete-state stochastic model for T cell signaling, that adopts a single molecule view of the antigen discrimination process. Our theory, which relies on modeling T cell receptor (TCR) phosphorylation events as discrete-state stochastic processes, suggests that the decision to activate or not to activate the immune response is controlled by the time to reach the stationary concentration of the TCR-ligand activated complex. Our result is consistent with the experimental observation that sometimes the T cell activation occurs even for very large ligand association rates. Mover, experimental evidence suggests that three characteristics are crucial to the function of T cells. These characteristics, which are speed, sensitivity and specificity, are described by our stochastic model.

The effect of obstacles in multi-site protein target search with DNA looping

Many fundamental biological processes are regulated by protein-DNA complexes called synaptosomes, which possess multiple interaction sites. Despite the critical importance of synaptosomes, the mechanisms of their formation are not well understood. Because of the multisite nature of participating proteins, it is widely believed that their search for specific sites on DNA involves the formation and breaking of DNA loops and sliding in the looped configurations. In reality, DNA in live cells is densely covered by other biological molecules that might interfere with the formation of synaptosomes. In this work, we developed a theoretical approach to evaluate the role of obstacles in the target search of multisite proteins when the formation of DNA loops and the sliding in looped configurations are possible. Our theoretical method is based on analysis of a discrete-state stochastic model that uses a master equations approach and extensive computer simulations. It is found that the obstacle slows down the search dynamics in the system when DNA loops are long-lived, but the effect is minimal for short-lived DNA loops. In addition, the relative positions of the target and the obstacle strongly influence the target search kinetics. Furthermore, the presence of the obstacle might increase the noise in the system. These observations are discussed using physical-chemical arguments. Our theoretical approach clarifies the molecular mechanisms of formation of protein-DNA complexes with multiple interactions sites.

Molecular search with conformational change: One-dimensional discrete-state stochastic model.

Molecular search phenomena are observed in a variety of chemical and biological systems. During the search, the participating particles frequently move in complex inhomogeneous environments with random transitions between different dynamic modes. To understand the mechanisms of molecular search with alternating dynamics, we investigate the search dynamics with stochastic transitions between two conformations in a one-dimensional discrete-state stochastic model. It is explicitly analyzed using the first-passage time probability method to obtain a full dynamic description of the search process. A general dynamic phase diagram is developed. It is found that there are several dynamic regimes in the molecular search with conformational transitions, and they are determined by the relative values of the relevant length scales in the system. Theoretical predictions are fully supported by Monte Carlo computer simulations.

Theoretical Investigations of the Role of Mutations in Dynamics of Kinesin Motor Proteins.

Motor proteins are active enzymatic molecules that are critically important for a variety of biological phenomena. It is known that some neurodegenerative diseases are caused by specific mutations in motor proteins that lead to their malfunctioning. Hereditary spastic paraplegia is one of such diseases, and it is associated with the mutations in the neuronal conventional kinesin gene, producing the decreased speed and processivity of this motor protein. Despite the importance of this problem, there is no clear understanding on the role of mutations in modifying dynamic properties of motor proteins. In this work, we investigate theoretically the molecular basis for negative effects of two specific mutations, N256S and R280S, on the dynamics of kinesin motor proteins. We hypothesize that these mutations might accelerate the adenosine triphosphate (ATP) release by increasing the probability of open conformations for the ATP-binding pocket. Our approach is based on the use of coarse-grained structure-based molecular dynamics simulations to analyze the conformational changes and chemical transitions in the kinesin molecule, which is also supplemented by investigation of a mesoscopic discrete-state stochastic model. Computer simulations suggest that mutations N256S and R280S can decrease the free energy difference between open and closed biochemical states, making the open conformation more stable and the ATP release faster, which is in agreement with our hypothesis. Furthermore, we show that in the case of N256S mutation, this effect is caused by disruption of interactions between α helix and switch I and loop L11 structural elements. Our computational results are qualitatively supported by the explicit analysis of the discrete-state stochastic model.

Mechanism of Genome Interrogation: How CRISPR RNA-Guided Cas9 Proteins Locate Specific Targets on DNA

The ability to precisely edit and modify a genome opens endless opportunities to investigate fundamental properties of living systems as well as to advance various medical techniques and bioengineering applications. This possibility is now close to reality due to a recent discovery of the adaptive bacterial immune system, which is based on clustered regularly interspaced short palindromic repeats (CRISPR)-associated proteins (Cas) that utilize RNA to find and cut the double-stranded DNA molecules at specific locations. Here we develop a quantitative theoretical approach to analyze the mechanism of target search on DNA by CRISPR RNA-guided Cas9 proteins, which is followed by a selective cleavage of nucleic acids. It is based on a discrete-state stochastic model that takes into account the most relevant physical-chemical processes in the system. Using a method of first-passage processes, a full dynamic description of the target search is presented. It is found that the location of specific sites on DNA by CRISPR Cas9 proteins is governed by binding first to protospacer adjacent motif sequences on DNA, which is followed by reversible transitions into DNA interrogation states. In addition, the search dynamics is strongly influenced by the off-target cutting. Our theoretical calculations allow us to explain the experimental observations and to give experimentally testable predictions. Thus, the presented theoretical model clarifies some molecular aspects of the genome interrogation by CRISPR RNA-guided Cas9 proteins.

On the Mechanism of Homology Search by RecA Protein Filaments

Genetic stability is a key factor in maintaining, survival, and reproduction of biological cells. It relies on many processes, but one of the most important is a homologous recombination, in which the repair of breaks in double-stranded DNA molecules is taking place with a help of several specific proteins. In bacteria, this task is accomplished by RecA proteins that are active as nucleoprotein filaments formed on single-stranded segments of DNA. A critical step in the homologous recombination is a search for a corresponding homologous region on DNA, which is called a homology search. Recent single-molecule experiments clarified some aspects of this process, but its molecular mechanisms remain not well understood. We developed a quantitative theoretical approach to analyze the homology search. It is based on a discrete-state stochastic model that takes into account the most relevant physical-chemical processes in the system. Using a method of first-passage processes, a full dynamic description of the homology search is presented. It is found that the search dynamics depends on the degree of extension of DNA molecules and on the size of RecA nucleoprotein filaments, in agreement with experimental single-molecule measurements of DNA pairing by RecA proteins. Our theoretical calculations, supported by extensive Monte Carlo computer simulations, provide a molecular description of the mechanisms of the homology search.

How Conformational Dynamics Influences the Protein Search for Targets on DNA

Protein search and association to specific target sequences on DNA is essential for all fundamental biological processes. The detailed qualitative and quantitative mechanism of fast target finding is still unknown in spite of numerous experimental and theoretical efforts. Particularly, the role of protein conformational fluctuations in the search dynamics remains uncovered. We extend developed earlier theoretical method to analyze how the conformational dynamic affects the process of finding the specific targets on DNA. Our approach is based on discrete-state stochastic model that takes into account main physical-chemical processes. This allows us to evaluate explicitly the protein search for the targets on DNA at different conditions. Our calculations show that conformational fluctuations might strongly affect the protein search dynamics. We discuss the contribution of the shift in the conformational equilibrium in the target search kinetics. We utilized extensive Monte Carlo computer simulations to validate our theoretical predictions.

Stopped-Flow Fluorescence Kinetic Study of Protein Sliding and Intersegment Transfer in the Target DNA Search Process

Kinetic characterizations of protein translocation on DNA are nontrivial because the simultaneous presence of multiple different mechanisms makes it difficult to extract the information specific to a particular translocation mechanism. In this study, we have developed new approaches for the kinetic investigations of proteins' sliding and intersegment transfer (also known as “direct transfer”) in the target DNA search process. Based on the analytical expression of the mean search time for the discrete-state stochastic model, we derived analytical forms of the apparent rate constant kapp for protein-target association in systems involving competitor DNA and the intersegment transfer mechanism. Our analytical forms of kapp facilitate the experimental determination of the kinetic rate constants for intersegment transfer and sliding in the target association process. Using stopped-flow fluorescence data for the target association kinetics along with the analytical forms of kapp, we have studied the translocation of the Egr-1 zinc-finger protein in the target DNA association process. Sliding was analyzed using the DNA-length-dependent kapp data. Using the dependence of kapp on the concentration of competitor DNA, we determined the second-order rate constant for intersegment transfer. Our results indicate that a major pathway in the target association process for the Egr-1 zinc-finger protein is the one involving intersegment transfer to a nonspecific site and the subsequent sliding to the target.

Dynamic properties of molecular motors in the divided-pathway model

The mechanisms of molecular motors transport are important for understanding multiple biological processes. Recent single-molecule experiments indicate that motor proteins myosin V moves along protein filaments via a complex biochemical pathway that consists of sequentially coupled linear and parallel two-chain segments. We investigate analytically the corresponding discrete-state stochastic divided-pathway model for molecular motors transport. Explicit expressions are obtained for velocities and dispersions. The dynamic properties of motor proteins in the divided-pathway model are compared with those in single-chain linear and parallel-pathway stochastic models. It is argued that modifying biochemical pathways has a strong effect on the dynamic properties, and it allows motor proteins to be more flexible in performing their biological functions.

A Stochastic Model of Oscillatory Blood Testosterone Levels

A continuous-time, discrete-state stochastic model of testosterone secretion in men is considered. Blood levels of testosterone in men fluctuate periodically with a period of 2-3 h. The deterministic model, on which the stochastic model considered here is based, is well studied and has been shown to have a globally stable fixed point. Thus, no sustained oscillations are possible in the deterministic case. However, the stochastic model does observe periodic, pulsatile behavior. This demonstrates how oscillations can occur due to a switching behavior dependent on the random degradation of testosterone molecules in the system. The Gillespie algorithm is used to simulate the hormone secretion model. Important parameters of the model are discussed and results from the model are compared to experimental observations.

Volatility dynamics under duration-dependent mixing

This paper proposes a discrete-state stochastic volatility model with duration-dependent mixing. The latter is directed by a high-order Markov chain with a sparse transition matrix. As in the standard first-order Markov switching (MS) model, this structure can capture turning points and shifts in volatility, due for example, to policy changes or news events. However, the duration-dependent Markov switching model (DDMS) can also exploit the persistence associated with volatility clustering. To evaluate the contribution of duration dependence, we compare with a benchmark Markov switching-ARCH (MS-ARCH) model. The empirical distribution generated by our proposed structure is assessed using interval forecasts and density forecasts. Implications for areas of the distribution relevant to risk management are also assessed.

Women, work, and identities

This paper uses the concept of identity to understand constancy and change in women's work histories. Identities are self-in-role meanings such as parent, worker, or professional. Identities are relational, hierarchical, and have consistency and motivational implications for behavior. From a life course perspective, we posit that identities both help organize and also change with the course of people's lives. We illustrate the organizing role of identities with a continuous-time discrete-state stochastic model of women's movement between job-related identities (employed but not as part of a career and employed in a career) and less-than-full-time employment. The model considers the rates of transitions into and out of identity states as functions of fixed and changing personal resources, changes in stage of the family life cycle, rewards and opportunities associated with the present job and the career line, and several forms of duration dependence (time spent in the state). Data come from life histories of a Washington state sample of women, studied at age 30 in 1979 and 13 years earlier in 1966 ( N = 2536). The analyses suggest work identities operate largely as hypothesized and that the concept has potential for understanding work histories.