Gene expression in a cell entails random reaction events occurring over disparate time scales. Thus, molecular noise that often results in phenotypic and population-dynamic consequences sets a fundamental limit to biochemical signaling. While there have been numerous studies correlating the architecture of cellular reaction networks with noise tolerance, only a limited effort has been made to understand the dynamic role of protein-protein interactions. Here we have developed a fully stochastic model for the positive feedback control of a single gene, as well as a pair of genes (toggle switch), integrating quantitative results from previous in vivo and in vitro studies. We find that the overall noise-level is reduced and the frequency content of the noise is dramatically shifted to the physiologically irrelevant high-frequency regime in the presence of protein dimerization. This is independent of the choice of monomer or dimer as transcription factor and persists throughout the multiple model topologies considered. For the toggle switch, we additionally find that the presence of a protein dimer, either homodimer or heterodimer, may significantly reduce its random switching rate. Hence, the dimer promotes the robust function of bistable switches by preventing the uninduced (induced) state from randomly being induced (uninduced). The specific binding between regulatory proteins provides a buffer that may prevent the propagation of fluctuations in genetic activity. The capacity of the buffer is a non-monotonic function of association-dissociation rates. Since the protein oligomerization per se does not require extra protein components to be expressed, it provides a basis for the rapid control of intrinsic or extrinsic noise.
Genetic interactions have been widely used to define functional relationships between proteins and pathways. In this study, we demonstrated that yeast synthetic lethal genetic interactions can be explained by the genetic interactions between domains of those proteins. The domain genetic interactions rarely overlap with the domain physical interactions from iPfam database and provide a complementary view about domain relationships. Moreover, we found that domains in multidomain yeast proteins contribute to their genetic interactions differently. The domain genetic interactions help more precise…
Allostery is a form of protein regulation, where ligands that bind sites located apart from the active site can modify the activity of the protein. The molecular mechanisms of allostery have been extensively studied, because allosteric sites are less conserved than active sites, and drugs targeting them are more specific than drugs binding the active sites. Here we quantify the importance of allostery in genetic disease. We show that 1) known allosteric proteins are central in disease networks, and contribute to genetic disease and comorbidities much more than non-allosteric proteins, in many …
Theoretical physics is used for a toy model of molecular biology to assess conditions that lead to the edge of chaos (EOC) in a network of biomolecules. Results can enhance our ability to understand complex diseases and their treatment or cure.
We fit the parameters of a differential equations model describing the production of gap gene proteins Hunchback and Knirps along the antero-posterior axis of the embryo of \emph{Drosophila}. As initial data for the differential equations model, we take the antero-posterior distribution of the proteins Bicoid, Hunchback and Tailless at the beginning of cleavage cycle 14. We calibrate and validate the model with experimental data using single- and multi-objective evolutionary optimization techniques. In the multi-objective optimization technique, we compute the associated Pareto fronts. We anal…
The so-called Mixed Feedback Loop (MFL) is a small two-gene network where protein A regulates the transcription of protein B and the two proteins form a heterodimer. It has been found to be statistically over-represented in statistical analyses of gene and protein interaction databases and to lie at the core of several computer-generated genetic networks. Here, we propose and mathematically study a model of the MFL and show that, by itself, it can serve both as a bistable switch and as a clock (an oscillator) depending on kinetic parameters. The MFL phase diagram as well as a detailed descript…
The functions of most genetic circuits require sufficient degrees of cooperativity in the circuit components. While mechanisms of cooperativity have been studied most extensively in the context of transcriptional initiation control, cooperativity from other processes involved in the operation of the circuits can also play important roles. In this study, we examine a simple kinetic source of cooperativity stemming from the nonlinear degradation of multimeric proteins. Ample experimental evidence suggests that protein subunits can degrade less rapidly when associated in multimeric complexes, an …
We present a general methodology in order to build mathematical models of genetic regulatory networks. This approach is based on the mass action law and on the Jacob and Monod operon model. The mathematical models are built symbolically by the \emph{Mathematica} software package \emph{GeneticNetworks}. This package accepts as input the interaction graphs of the transcriptional activators and repressors and, as output, gives the mathematical model in the form of a system of ordinary differential equations. All the relevant biological parameters are chosen automatically by the software. Within t…
In many stochastic simulations of biochemical reaction networks, it is desirable to ``coarse-grain'' the reaction set, removing fast reactions while retaining the correct system dynamics. Various coarse-graining methods have been proposed, but it remains unclear which methods are reliable and which reactions can safely be eliminated. We address these issues for a model gene regulatory network that is particularly sensitive to dynamical fluctuations: a bistable genetic switch. We remove protein-DNA and/or protein-protein association-dissociation reactions from the reaction set, using various co…
Background The theoretical requirements for a genetic code were well defined and modeled by George Gamow and Francis Crick in the 50-es. Their models failed. However the valid Genetic Code, provided by Nirenberg and Matthaei in 1961, ignores many theoretical requirements for a perfect Code. Something is simply missing from the canonical Code. Results The 3x redundancy of the Genetic code is usually explained as a necessity to increase the resistance of the mutation resistance of the genetic information. However it has many additional roles. 1.) It has a periodical structure which corresponds t…
The ability to learn and respond to recurrent events depends on the capacity to remember transient biological signals received in the past. Moreover, it may be desirable to remember or ignore these transient signals conditioned upon other signals that are active at specific points in time or in unique environments. Here, we propose a simple genetic circuit in bacteria that is capable of conditionally memorizing a signal in the form of a transcription factor concentration. The circuit behaves similarly to a "data latch" in an electronic circuit, i.e. it reads and stores an input signal only whe…
Auto-regulation, a process wherein a protein negatively regulates its own production, is a common motif in gene expression networks. Negative feedback in gene expression plays a critical role in buffering intracellular fluctuations in protein concentrations around optimal value. Due to the nonlinearities present in these feedbacks, moment dynamics are typically not closed, in the sense that the time derivative of the lower-order statistical moments of the protein copy number depends on high-order moments. Moment equations are closed by expressing higher-order moments as nonlinear functions of …
Cells receive a wide variety of cellular and environmental signals, which must be processed combinatorially to generate specific and timely genetic responses. We present here a theoretical study on the combinatorial control and integration of transcription signals, with the finding that cis-regulatory systems with specific protein-DNA interaction and glue-like protein-protein interactions, supplemented by distal activation or repression mechanisms, have the capability to execute a wide range of control functions encoded in the regulatory DNA sequences. Using a quantitative model based on the w…
We investigate the dynamics of the heterodimer autorepression loop (HAL), a small genetic module in which a protein A acts as an auto-repressor and binds to a second protein B to form a AB dimer. For suitable values of the rate constants the HAL produces pulses of A alternating with pulses of B. By means of analytical and numerical calculations, we show that the duration of A-pulses is extremely robust against variation of the rate constants while the duration of the B-pulses can be flexibly adjusted. The HAL is thus a minimal genetic module generating robust pulses with tunable duration an in…
We study the noise characteristics of stochastic oscillations in protein number dynamics of simple genetic oscillatory systems. Using the three-component negative feedback transcription regulatory system called the repressilator as a prototypical example, we quantify the degree of fluctuations in oscillation periods and amplitudes, as well as the noise propagation along the regulatory cascade in the stable oscillation regime via dynamic Monte Carlo simulations. For the single protein-species level, the fluctuation in the oscillation amplitudes is found to be larger than that of the oscillation…
Gene products (RNAs, proteins) often occur at low molecular counts inside individual cells, and hence are subject to considerable random fluctuations (noise) in copy number over time. Not surprisingly, cells encode diverse regulatory mechanisms to buffer noise. One such mechanism is the incoherent feedforward circuit. We analyze a simplistic version of this circuit, where an upstream regulator X affects both the production and degradation of a protein Y. Thus, any random increase in X's copy numbers would increase both production and degradation, keeping Y levels unchanged. To study its stocha…
The phage $λ$ infection of an \textit{E. coli} cell has become a paradigm for understanding the molecular processes involved in gene expression and cell signaling. This system provides an example of a genetic switch, as cells with identical DNA choose either of two cell cycles: a lysogenic cycle, in which the phage genome is incorporated into the host and copied by the host; or a lytic cycle, resulting in the death of the cell and a burst of viruses. The robustness of this switch is remarkable; although the first stages of the lysogenic and lytic cycles are identical, a lysogen rarely spontane…
The digital nature of genes combined with the associated low copy numbers of proteins regulating them is a significant source of stochasticity, which affects the phase of biochemical oscillations. We provide a theoretical framework for understanding the dephasing evolution of genetic oscillations by combining the phenomenological stochastic limit cycle dynamics and the discrete Markov state models that describe the genetic oscillations. Through simulations of the realistic model of the NFκB/IκB network we illustrate the dephasing phenomena which are important for reconciling single cell and po…
Auto-regulatory feedback loops are one of the most common network motifs. A wide variety of stochastic models have been constructed to understand how the fluctuations in protein numbers in these loops are influenced by the kinetic parameters of the main biochemical steps. These models differ according to (i) which sub-cellular processes are explicitly modelled; (ii) the modelling methodology employed (discrete, continuous or hybrid); (iii) whether they can be analytically solved for the steady-state distribution of protein numbers. We discuss the assumptions and properties of the main models i…
High-gain nonlinear observers occur in the nonlinear automatic control theory and are in standard usage in chemical engineering processes. We apply such a type of analysis in the context of a very simple one-gene regulation circuit. In general, an observer combines an analytical differential-equation-based model with partial measurement of the system in order to estimate the non-measured state variables. We use one of the simplest observers, that of Gauthier et al., which is a copy of the original system plus a correction term which is easy to calculate. For the illustration of this procedure,…
Numerous studies have demonstrated the important role of noise in the dynamical behaviour of a complex system. The most probable trajectories of nonlinear systems under the influence of Gaussian noise have recently been studied already. However, there has been only a few works that examine how most probable trajectories in the two-dimensional system (MeKS network) are influenced under non-Gaussian stable Lévy noise. Therefore, we discuss the most probable trajectories of a two-dimensional model depicting the competence behaviour in B. subtilis under the influence of stable Lévy noise. On the b…
A self-regulatory genetic circuit, where a protein acts as a positive regulator of its own production, is known to be a simplest form of biological network with a positive feedback loop. Although at least three components, DNA, RNA, and the protein, are required to form such a circuit, the stability analysis of fixed points of the self-regulatory circuit has been performed only after reducing the system into to a two-component system consisting of RNA and protein only, assuming a fast equilibration of the DNA component. Here, the stability of fixed points of the three-component positive feedba…
Cells use genetic switches to shift between alternate gene expression states, e.g., to adapt to new environments or to follow a developmental pathway. Here, we study the dynamics of switching in a generic-feedback on/off switch. Unlike protein-only models, we explicitly account for stochastic fluctuations of mRNA, which have a dramatic impact on switch dynamics. Employing the WKB theory to treat the underlying chemical master equations, we obtain accurate results for the quasi-stationary distributions of mRNA and protein copy numbers and for the mean switching time, starting from either state.…
Cells are known to utilize biochemical noise to probabilistically switch between distinct gene expression states. We demonstrate that such noise-driven switching is dominated by tails of probability distributions and is therefore exponentially sensitive to changes in physiological parameters such as transcription and translation rates. However, provided mRNA lifetimes are short, switching can still be accurately simulated using protein-only models of gene expression. Exponential sensitivity limits the robustness of noise-driven switching, suggesting cells may use other mechanisms in order to s…
From a coarse-grained perspective the motif of a self-activating species, activating a second species which acts as its own repressor, is widely found in biological systems, in particular in genetic systems with inherent oscillatory behavior. Here we consider a specific realization of this motif as a genetic circuit, in which genes are described as directly producing proteins, leaving out the intermediate step of mRNA production. We focus on the effect that inherent time scales on the underlying fine-grained scale can have on the bifurcation patterns on a coarser scale in time. Time scales are…
Living cells must control the reading out or "expression" of information encoded in their genomes, and this regulation often is mediated by transcription factors--proteins that bind to DNA and either enhance or repress the expression of nearby genes. But the expression of transcription factor proteins is itself regulated, and many transcription factors regulate their own expression in addition to responding to other input signals. Here we analyze the simplest of such self-regulatory circuits, asking how parameters can be chosen to optimize information transmission from inputs to outputs in the…
We study by mean-field analysis and stochastic simulations chemical models for genetic toggle switches formed from pairs of genes that mutually repress each other. In order to determine the stability of the genetic switches, we make a connection with reactive flux theory and transition state theory. The switch stability is characterised by a well defined lifetime $τ$. We find that $τ$ grows exponentially with the mean number $\Nmean$ of transcription factor molecules involved in the switching. In the regime accessible to direct numerical simulations, the growth law is well characterised by $τ\…
Stochastic kinetic models of genetic expression are able to describe protein fluctuations. A comparative study of the canonical and a feedback model is given here by using stochastic simulation methods. The feedback model is skeleton model implementation of the circular gene hypothesis, which suggests the interaction between the synthesis and degradation of mRNA. Qualitative and quantitative changes in the shape and in the numerical characteristics of the stationary distributions suggest that more combined experimental and theoretical studies should be done to uncover the details of the kineti…
In order to survive, reproduce and (in multicellular organisms) differentiate, cells must control the concentrations of the myriad different proteins that are encoded in the genome. The precision of this control is limited by the inevitable randomness of individual molecular events. Here we explore how cells can maximize their control power in the presence of these physical limits; formally, we solve the theoretical problem of maximizing the information transferred from inputs to outputs when the number of available molecules is held fixed. We start with the simplest version of the problem, in…
A wide range of organisms use circadian clocks to keep internal sense of daily time and regulate their behavior accordingly. Most of these clocks use intracellular genetic networks based on positive and negative regulatory elements. The integration of these "circuits" at the cellular level imposes strong constraints on their functioning and design. Here we study a recently proposed model [N. Barkai and S. Leibler, Nature, 403:267--268, 2000] that incorporates just the essential elements found experimentally. We show that this type of oscillator is driven mainly by two elements: the concentrati…
Two synthetic gene circuits -- the genetic toggle switch and the repressilator -- are analyzed quantitatively and discussed in the context of an educational module on gene circuits and feedback that constitutes the final topic of a year-long introductory physics sequence, aimed at biology and premedical undergraduate students. The genetic toggle switch consists of two genes, each of whose protein product represses the other's expression, while the repressilator consists of three genes, each of whose protein product represses the next gene's expression. Analytic, numerical, and electronic treat…
We discuss recent developments in the modeling of negative autoregulated genetic networks. In particular, we consider the temporal evolution of the population of mRNA and proteins in simple networks using rate equations. In the limit of low copy numbers, fluctuation effects become significant and more adequate modeling is then achieved using the master equation formalism. The analogy between regulatory gene networks and chemical reaction networks on dust grains in the interstellar medium is discussed. The analysis and simulation of complex reaction networks are also considered.
In high-throughput functional genomic screens, each gene product is commonly assumed to exhibit a singular biological function within a defined protein complex or pathway. In practice, a single gene perturbation may induce multiple cascading functional outcomes, a genetic principle known as pleiotropy. Here, we model pleiotropy in fitness screen collections by representing each gene perturbation as the sum of multiple perturbations of biological functions, each harboring independent fitness effects inferred empirically from the data. Our approach ('Webster') recovered pleiotropic functions for…
Control theory has seen recently impactful applications in network science, especially in connections with applications in network medicine. A key topic of research is that of finding minimal external interventions that offer control over the dynamics of a given network, a problem known as network controllability. We propose in this article a new solution for this problem based on genetic algorithms. We tailor our solution for applications in computational drug repurposing, seeking to maximise its use of FDA-approved drug targets in a given disease-specific protein-protein interaction network.…
Negative and positive transcriptional feedback loops are present in natural and synthetic genetic oscillators. A single gene with negative transcriptional feedback needs a time delay and sufficiently strong nonlinearity in the transmission of the feedback signal in order to produce biochemical rhythms. A single gene with only positive transcriptional feedback does not produce oscillations. Here, we demonstrate that this single-gene network in conjunction with a simple negative interaction can also easily produce rhythms. We examine a model comprised of two well-differentiated parts. The first …
Changes in a cell's external or internal conditions are usually reflected in the concentrations of the relevant transcription factors. These proteins in turn modulate the expression levels of the genes under their control and sometimes need to perform non-trivial computations that integrate several inputs and affect multiple genes. At the same time, the activities of the regulated genes would fluctuate even if the inputs were held fixed, as a consequence of the intrinsic noise in the system, and such noise must fundamentally limit the reliability of any genetic computation. Here we use informa…
A description for regulatory genetic network based on generalized potential energy is constructed. The potential energy is derived from the steady state solution of linearized Fokker-Plank equation, and the result is shown to be equivalent to the system of coupled oscillators. The correspondence between the quantities from the mechanical picture and the steady-state fluctuations is established. Explicit calculation is given for auto-regulatory networks in which, the force constant associated with the degree of protein is very weak. Negative feedback not only suppresses the fluctuations but als…
Genetic circuits need a cellular environment to operate in, which naturally couples the circuit function with the overall functionality of gene regulatory network. To execute their functions all gene circuits draw resources in the form of RNA polymerases, ribosomes, and tRNAs. Recent experiments pointed out that the role of resource competition on synthetic circuit outputs could be immense. However, the effect of complexity of the circuit architecture on resource sharing dynamics is yet unexplored. In this paper, we employ mathematical modelling and in-silico experiments to identify the source…
We study a biological autoregulation process, involving a protein that enhances its own transcription, in a parameter region where bistability would be present in the absence of fluctuations. We calculate the rate of fluctuation-induced rare transitions between locally-stable states using a path integral formulation and Master and Chapman-Kolmogorov equations. As in simpler models for rare transitions, the rate has the form of the exponential of a quantity $S_0$ (a "barrier") multiplied by a prefactor $η$. We calculate $S_0$ and $η$ first in the bursting limit (where the ratio $γ$ of the prote…
Stochastic dynamics of chemical reactions in a mutually repressing two-gene circuit is numerically simulated. The circuit has a rich variety of different states when the kinetic change of DNA status is slow. The stochastic switching transition between those states are compared with the theoretical estimation of the switching rate derived from the idea similar to the transition state theory. Even though the circuit is kept far from equilibrium, the method gives a consistent explanation of the switching kinetics for a wide range of parameters. The transition state theory-like estimation, however…
In this review, we give an introduction to the structural and functional properties of the biological networks. We focus on three major themes: topology of complex biological networks like the metabolic and protein-protein interaction networks, nonlinear dynamics in gene regulatory networks and in particular the design of synthetic genetic networks using the concepts and techniques of nonlinear physics and lastly the effect of stochasticity on the dynamics. The examples chosen illustrate the usefulness of interdisciplinary approaches in the study of biological networks.
