First passage and cooperativity of queuing kinetics

Efficient Low-Order Approximation of First-Passage Time Distributions

We consider the problem of computing first-passage time distributions for reaction processes modeled by master equations. We show that this generally intractable class of problems is equivalent to a sequential Bayesian inference problem for an auxiliary observation process. The solution can be approximated efficiently by solving a closed set of coupled ordinary differential equations (for the low-order moments of the process) whose size scales with the number of species. We apply it to an epidemic model and a trimerization process and show good agreement with stochastic simulations.

Kinetics of self-assembly via facilitated diffusion: Formation of the transcription complex

We present an analytically solvable model for self-assembly of a molecular complex on a filament. The process is driven by a seed molecule that undergoes facilitated diffusion, which is a search strategy that combines diffusion in three dimensions and one dimension. Our study is motivated by single-molecule-level observations revealing the dynamics of transcription factors that bind to the deoxyribonucleic acid at early stages of transcription. We calculate the probability that a complex made up of a given number of molecules is completely formed, as well as the distribution of completion times, upon the binding of a seed molecule at a target site on the filament (without explicitly modeling the three-dimensional diffusion that precedes binding). We compare two different mechanisms of assembly where molecules bind in sequential and random order. Our results indicate that while the probability of completion is greater for random binding, the completion time scales exponentially with the size of the complex; in contrast, it scales as a power law or slower for sequential binding, asymptotically. Furthermore, we provide model predictions for the dissociation and residence times of the seed molecule, which are observables accessible in single-molecule tracking experiments.

Assembly of the transcription machinery: ordered and stable, random and dynamic, or both?

The assembly of the transcription machinery is a key step in gene activation, but even basic details of this process remain unclear. Here we discuss the apparent discrepancy between the classic sequential assembly model based mostly on biochemistry and an emerging dynamic assembly model based mostly on fluorescence microscopy. The former model favors a stable transcription complex with subunits that cooperatively assemble in order, whereas the latter model favors an unstable complex with subunits that may assemble more randomly. To confront this apparent discrepancy, we review the merits and drawbacks of the different experimental approaches and list potential biasing factors that could be responsible for the different interpretations of assembly. We then discuss how these biases might be overcome in the future with improved experiments or new techniques. Finally, we discuss how kinetic models for assembly may help resolve the ordered and stable vs. random and dynamic assembly debate.

Calculating transcription factor binding maps for chromatin

Current high-throughput experiments already generate enough data for retrieving the DNA sequence-dependent binding affinities of transcription factors (TF) and other chromosomal proteins throughout the complete genome. However, the reverse task of calculating binding maps in a chromatin context for a given set of concentrations and TF affinities appears to be even more challenging and computationally demanding. The problem can be addressed by considering the DNA sequence as a one-dimensional lattice with units of one or more base pairs. To calculate protein occupancies in chromatin, one needs to consider the competition of TF and histone octamers for binding sites as well as the partial unwrapping of nucleosomal DNA. Here, we consider five different classes of algorithms to compute binding maps that include the binary variable, combinatorial, sequence generating function, transfer matrix and dynamic programming approaches. The calculation time of the binary variable algorithm scales exponentially with DNA length, which limits its use to the analysis of very small genomic regions. For regulatory regions with many overlapping binding sites, potentially applicable algorithms reduce either to the transfer matrix or dynamic programming approach. In addition to the recently proposed transfer matrix formalism for TF access to the nucleosomal organized DNA, we develop here a dynamic programming algorithm that accounts for this feature. In the absence of nucleosomes, dynamic programming outperforms the transfer matrix approach, but the latter is faster when nucleosome unwrapping has to be considered. Strategies are discussed that could further facilitate calculations to allow computing genome-wide TF binding maps.

Coarsening and accelerated equilibration in mass-conserving heterogeneous nucleation

We propose a model of mass-conserving heterogeneous nucleation to describe the dynamics of ligand-receptor binding in closed cellular compartments. When the ligand dissociation rate is small, competition among receptors for free ligands gives rise to two very different long-time ligand-receptor cluster-size distributions. Cluster sizes first plateau to a long-lived, initial-condition-dependent, "metastable" distribution, and coarsen only much later to a qualitatively different equilibrium one. Surprisingly, we also find parameters for which a very special subset of clusters have equal metastable and equilibrium sizes, appearing to equilibrate much faster than the rest. Our results provide a quantitative framework for ligand-binding kinetics and suggest a mechanism by which different clusters can approach their equilibrium sizes in unexpected ways.

Diffusion-dependent mechanisms of receptor engagement and viral entry

Enveloped viruses attach to host cells by binding to receptors on the cell surface. For many viruses, entry occurs via membrane fusion after a sufficient number of receptors have engaged ligand proteins on the virion. Under conditions where the cell surface receptor densities are low, recruitment of receptors may be limited by diffusion rather than by receptor-ligand interactions. We present a receptor-binding model that includes the effects of receptor availability at the viral binding site. The receptor binding and unbinding kinetics are coupled to receptor diffusion across the cell membrane. We find numerical solutions to our model and analyze the viral entry probabilities and the mean times to entry as functions of receptor concentration, receptor diffusivity, receptor binding stoichiometry, receptor detachment rates, and virus degradation/detachment rates. We also show how entry probabilities and times differ when receptors bind randomly or sequentially to the binding sites on the viral glycoprotein spikes. Our results provide general insight into the biophysical transport mechanisms that may arise in viral attachment and entry.

The simplicity of completion time distributions for common complex biochemical processes

Biochemical processes typically involve huge numbers of individual reversible steps, each with its own dynamical rate constants. For example, kinetic proofreading processes rely upon numerous sequential reactions in order to guarantee the precise construction of specific macromolecules. In this work, we study the transient properties of such systems and fully characterize their first passage (completion) time distributions. In particular, we provide explicit expressions for the mean and the variance of the completion time for a kinetic proofreading process and computational analyses for more complicated biochemical systems. We find that, for a wide range of parameters, as the system size grows, the completion time behavior simplifies: it becomes either deterministic or exponentially distributed, with a very narrow transition between the two regimes. In both regimes, the dynamical complexity of the full system is trivial compared to its apparent structural complexity. Similar simplicity is likely to arise in the dynamics of many complex multistep biochemical processes. In particular, these findings suggest not only that one may not be able to understand individual elementary reactions from macroscopic observations, but also that such an understanding may be unnecessary.

Effects of jamming on nonequilibrium transport times in nanochannels

Many biological channels perform highly selective transport without direct input of metabolic energy and without transitions from a "closed" to an "open" state during transport. Mechanisms of selectivity of such channels serve as an inspiration for creation of artificial nanomolecular sorting devices and biosensors. To elucidate the transport mechanisms, it is important to understand the transport on the single molecule level in the experimentally relevant regime when multiple particles are crowded in the channel. In this Letter we analyze the effects of interparticle crowding on the nonequilibrium transport times through a finite-length channel by means of analytical theory and computer simulations.

Mechanisms of receptor/coreceptor-mediated entry of enveloped viruses

Enveloped viruses enter host cells either through endocytosis, or by direct fusion of the viral envelope and the membrane of the host cell. However, some viruses, such as HIV-1, HSV-1, and Epstein-Barr can enter a cell through either mechanism, with the choice of pathway often a function of the ambient physical chemical conditions, such as temperature and pH. We develop a stochastic model that describes the entry process at the level of binding of viral glycoprotein spikes to cell membrane receptors and coreceptors. In our model, receptors attach the cell membrane to the viral membrane, while subsequent binding of coreceptors enables fusion. The model quantifies the competition between fusion and endocytotic entry pathways. Relative probabilities for each pathway are computed numerically, as well as analytically in the high viral spike density limit. We delineate parameter regimes in which fusion or endocytosis is dominant. These parameters are related to measurable and potentially controllable quantities such as membrane bending rigidity and receptor, coreceptor, and viral spike densities. Experimental implications of our mechanistic hypotheses are proposed and discussed.

Stochastic entry of enveloped viruses: fusion versus endocytosis

Infection by membrane-enveloped viruses requires the binding of receptors on the target cell membrane to glycoproteins, or "spikes," on the viral membrane. The initial entry mechanism is usually classified as fusogenic or endocytotic. However, binding of viral spikes to cell surface receptors not only initiates the viral adhesion and the wrapping process necessary for internalization, but can simultaneously initiate direct fusion with the cell membrane. Both fusion and internalization have been observed to be viable pathways for many viruses. We develop a stochastic model for viral entry that incorporates a competition between receptor-mediated fusion and endocytosis. The relative probabilities of fusion and endocytosis of a virus particle initially nonspecifically adsorbed on the host cell membrane are computed as functions of receptor concentration, binding strength, and number of spikes. We find different parameter regimes where the entry pathway probabilities can be analytically expressed. Experimental tests of our mechanistic hypotheses are proposed and discussed.