Khem Raj Ghusinga

First-passage time calculations for a gene expression model

The stochastic nature of gene expression can lead to significant cell-to-cell variability in the time at which a certain protein level is attained. This is reflected in the timing of cellular events triggering at critical protein thresholds as well. A problem of interest is to understand how cells regulate gene expression to ensure precise timing of important events. To this end, we consider a gene expression model assuming constitutive expression in translation bursts. We also assume the proteins to be stable. The event timing is formulated as a first-passage time (FPT) problem and stochasticity in FPT for this model is quantified. We also investigate the effect of auto-regulation, a control mechanism often present in cells, on the stochasticity of FPT. In particular, we ask: given FPT threshold of proteins and mean FPT, what form of auto-regulation minimizes variance in FPT? Our results show that the objective is best achieved by having no auto-regulation. Moreover, a smaller mean burst size would result into lower stochasticity. We discuss our results in context of lysis time of E. coli cells infected by a λ phage virus. An optimal lysis time provides evolutionary advantage to λ phage, suggesting a possible regulation to minimize its stochasticity. Our results are consistent with previous studies showing there is no auto-regulation of the protein responsible for lysis. Moreover, congruent with experimental evidences, our analysis predicts that the expression of the lysis protein should have a small burst size.

First-passage time approach to controlling noise in the timing of intracellular events.

Significance Understanding how randomness in the timing of intracellular events is buffered has important consequences for diverse cellular processes, where precision is required for proper functioning. To investigate event timing in noisy biochemical systems, we develop a first-passage time framework in which an event is triggered when a regulatory protein accumulates up to a critical level. Formulas quantifying event-timing fluctuations in stochastic models of protein synthesis with feedback regulation are developed. Formulas shed counterintuitive insights into regulatory mechanisms essential for scheduling an event at a precise time with minimal fluctuations. These results uncover various features in the biochemical pathways used by phages to lyse individually infected bacterial cells at an optimal time, despite stochastic expression of lysis proteins. In the noisy cellular environment, gene products are subject to inherent random fluctuations in copy numbers over time. How cells ensure precision in the timing of key intracellular events despite such stochasticity is an intriguing fundamental problem. We formulate event timing as a first-passage time problem, where an event is triggered when the level of a protein crosses a critical threshold for the first time. Analytical calculations are performed for the first-passage time distribution in stochastic models of gene expression. Derivation of these formulas motivates an interesting question: Is there an optimal feedback strategy to regulate the synthesis of a protein to ensure that an event will occur at a precise time, while minimizing deviations or noise about the mean? Counterintuitively, results show that for a stable long-lived protein, the optimal strategy is to express the protein at a constant rate without any feedback regulation, and any form of feedback (positive, negative, or any combination of them) will always amplify noise in event timing. In contrast, a positive feedback mechanism provides the highest precision in timing for an unstable protein. These theoretical results explain recent experimental observations of single-cell lysis times in bacteriophage λ. Here, lysis of an infected bacterial cell is orchestrated by the expression and accumulation of a stable λ protein up to a threshold, and precision in timing is achieved via feedforward rather than feedback control. Our results have broad implications for diverse cellular processes that rely on precise temporal triggering of events.

Exact lower and upper bounds on stationary moments in stochastic biochemical systems

In the stochastic description of biochemical reaction systems, the time evolution of statistical moments for species population counts is described by a linear dynamical system. However, except for some ideal cases (such as zero- and first-order reaction kinetics), the moment dynamics is underdetermined as lower-order moments depend upon higher-order moments. Here, we propose a novel method to find exact lower and upper bounds on stationary moments for a given arbitrary system of biochemical reactions. The method exploits the fact that statistical moments of any positive-valued random variable must satisfy some constraints that are compactly represented through the positive semidefiniteness of moment matrices. Our analysis shows that solving moment equations at steady state in conjunction with constraints on moment matrices provides exact lower and upper bounds on the moments. These results are illustrated by three different examples-the commonly used logistic growth model, stochastic gene expression with auto-regulation and an activator-repressor gene network motif. Interestingly, in all cases the accuracy of the bounds is shown to improve as moment equations are expanded to include higher-order moments. Our results provide avenues for development of approximation methods that provide explicit bounds on moments for nonlinear stochastic systems that are otherwise analytically intractable.

Optimal auto-regulation to minimize first-passage time variability in protein level

The timing of cellular events is inherently random because of the probabilistic nature of gene expression. Yet cells manage to have precise timing of important events. Here, we study how gene expression could possibly be regulated to precisely schedule timing of an event around a given time. Event timing is modeled as the first-passage time (FPT) for a proteins level to cross a critical threshold. Considering auto-regulation as a possible regulatory mechanism, we investigate what form of auto-regulation would lead to minimum stochasticity in FPT around a fixed time. We formulate a stochastic gene expression model and show that under certain assumptions, it reduces to a birth-death process. Our results show that when the death rate is zero, the objective is best achieved when all of the birth rates are equal. On the contrary, when the death rate is non-zero, the optimal birth rates are not equal. In terms of the gene expression model, these results illustrate that when protein does not degrade, stochasticity in FPT around a given time is minimized when there is no auto-regulation of its expression. However, when the protein degrades, some form of auto-regulation is required to achieve this. These results are consistent with experimental findings for the lysis time stochasticity in λ phage.

Theoretical predictions on the first-passage time for a gene expression model

Expression of a gene is inherently random, leading to variability in a proteins level across a population of cells with same genetic information and environment. Another consequence of this is the cell-to-cell variability in the time at which a certain protein level is achieved inside individual cells. In this work, we model such times using the first-passage time (FPT) framework. Gene expression is modeled in translation bursts wherein each mRNA molecule arrives as per a Poisson process, produces a geometrically distributed burst of protein molecules and degrades instantaneously. Also, the proteins are assumed to degrade as well. The FPT probability density function and statistical moments are determined for this model. In addition, the effects of change in model parameters (transcription rate, mean translation burst size, FPT threshold) on the mean and noise (quantified as the coefficient of variation squared) of FPT are studied. Our analysis shows that the mean FPT increases by increasing the FPT threshold or decreasing the protein production by a lower mean burst size or a lower transcription rate. The noise properties, however, show non-trivial pattern: a U-shape behavior is seen with respect to change in mean burst size or FPT threshold whereas a monotonous trend is observed for change in transcription rate. Lastly, we also discuss how these predictions can possibly be tested via experiments on the lysis time of the bacterial virus bacteriophage λ.

Cell size control and gene expression homeostasis in single-cells

Growth of a cell and its subsequent division into daughters is a fundamental aspect of all cellular living systems. During these processes, how do individual cells correct size aberrations so that they do not grow abnormally large or small? How do cells ensure that the concentration of essential gene products are maintained at desired levels, in spite of dynamic/stochastic changes in cell size during growth and division? Both these questions have fascinated researchers for over a century. We review how advances in singe-cell technologies and measurements are providing unique insights into these questions across organisms from prokaryotes to human cells. More specifically, diverse strategies based on timing of cell-cycle events, regulating growth, and number of daughters are employed to maintain cell size homeostasis. Interestingly, size homeostasis often results in size optimality - proliferation of individual cells in a population is maximized at an optimal cell size. We further discuss how size-dependent expression or gene-replication timing can buffer concentration of a gene product from cell-to-cell size variations within a population. Finally, we speculate on an intriguing hypothesis that specific size control strategies may have evolved as a consequence of gene-product concentration homeostasis.

Noise Analysis in Biochemical Complex Formation

Several biological functions are carried out via complexes that are formed via multimerization of either a single species (homomers) or multiple species (heteromers). Given functional relevance of these complexes, it is arguably desired to maintain their level at a set point and minimize fluctuations around it. Here we consider two simple models of complex formation – one for homomer and another for heteromer of two species – and analyze how important model parameters affect the noise in complex level. In particular, we study effects of (i) sensitivity of the complex formation rate with respect to constituting species’ abundance, and (ii) relative stability of the complex as compared with that of the constituents. By employing an approximate moment analysis, we find that for a given steady state level, there is an optimal sensitivity that minimizes noise (quantified by fano-factor; variance/mean) in the complex level. Furthermore, the noise becomes smaller if the complex is less stable than its constituents. Finally, for the heteromer case, our findings show that noise is enhanced if the complex is comparatively more sensitive to one constituent. We briefly discuss implications of our result for general complex formation processes.

Effect of gene-expression bursts on stochastic timing of cellular events

Gene expression is inherently a noisy process which manifests as cell-to-cell variability in time evolution of proteins. Consequently, events that trigger at critical threshold levels of regulatory proteins exhibit stochasticity in their timing. An important contributor to the noise in gene expression is translation bursts phenomena, which corresponds to randomness in number of proteins produced in a single mRNA lifetime. Modeling timing of an event as a first-passage time (FPT) problem, we explore the effect of burst size distribution on event timing. Towards this end, the probability density function of FPT is computed for a gene expression model where the burst size is drawn from a generic non-negative distribution. Analytical formulas for FPT moments are provided in terms of known vectors and inverse of a known matrix. The effect of burst size distribution is investigated by looking at how the feedback regulation strategy that minimizes noise in timing around a given time deviates from the strategy when the burst size is deterministic. Interestingly, results show that the feedback strategy for deterministic burst case is quite robust to change in burst size distribution, and deviations from it are confined to about 20% of the optimal value. These findings facilitate an improved understanding of noise regulation in event timing.

Stochastic optimal control using semidefinite programming for moment dynamics

This paper presents a method to approximately solve stochastic optimal control problems in which the cost function and the system dynamics are polynomial. For stochastic systems with polynomial dynamics, the moments of the state can be expressed as a, possibly infinite, system of deterministic linear ordinary differential equations. By casting the problem as a deterministic control problem in moment space, semidefinite programming is used to find a lower bound on the optimal solution. The constraints in the semidefinite program are imposed by the ordinary differential equations for moment dynamics and semidefiniteness of the outer product of moments. From the solution to the semidefinite program, an approximate optimal control strategy can be constructed using a least squares method. In the linear quadratic case, the method gives an exact solution to the optimal control problem. In more complex problems, an infinite number of moment differential equations would be required to compute the optimal control law. In this case, we give a procedure to increase the size of the semidefinite program, leading to increasingly accurate approximations to the true optimal control strategy.

Controlling noise in the timing of intracellular events: A first-passage time approach

In the noisy cellular environment, gene products are subject to inherent random fluctuations in copy numbers over time. How cells ensure precision in the timing of key intracellular events, in spite of such stochasticity is an intriguing fundamental problem. We formulate event timing as a first-passage time problem, where an event is triggered when the level of a protein crosses a critical threshold for the first time. Novel analytical calculations are preformed for the first-passage time distribution in stochastic models of gene expression, including models with feedback regulation. Derivation of these formulas motivates an interesting question: is there an optimal feedback strategy to regulate the synthesis of a protein to ensure that an event will occur at a precise time, while minimizing deviations or noise about the mean. Counter-intuitively, results show that for a stable long-lived protein, the optimal strategy is to express the protein at a constant rate without any feedback regulation, and any form of feedback (positive, negative or any combination of them) will always amplify noise in event timing. In contrast, a positive feedback mechanism provides the highest precision in timing for an unstable protein. These theoretical results explain recent experimental observations of single-cell lysis times in bacteriophage λ. Here, lysis of an infected bacterial cell is orchestrated by the expression and accumulation of a stable λ protein up to a threshold, and precision in timing is achieved via feedforward, rather than feedback control. Our results have broad implications for diverse cellular processes that rely on precise temporal triggering of events.