Signatures of combinatorial regulation in intrinsic biological noise

Abstract

Gene expression is controlled by the action of transcription factors that bind to DNA and influence the rate at which a gene is transcribed. The quantitative mapping between the regulator concentrations and the output of the gene is known as the cisregulatory input function (CRIF). Here, we show how the CRIF shapes the form of the joint probability distribution of molecular copy numbers of the regulators and the product of a gene. Namely, we derive a class of fluctuation-based relations that relate the moments of the distribution to the derivatives of the CRIF. These relations are useful because they enable statistics of naturally arising cell-to-cell variations in molecular copy numbers to substitute for traditional manipulations for probing regulatory mechanisms. We demonstrate that these relations can distinguish super-and subadditive gene regulatory scenarios (molecular analogs of AND and OR logic operations) in simulations that faithfully represent bacterial gene expression. Applications and extensions to other regulatory scenarios are discussed.gene expression | mathematical modeling | noise analysis inference | flow cytometry T ranscription factors regulate the expression of genes by binding to specific sites on the DNA that are typically spatially close to, sometimes even in, sequences that code for proteins. A group of such binding sites is collectively known as a cis-regulatory region. When a gene processes the effects of multiple transcription factors, one can view it as performing a computation. The inputs are the occupancies of the binding sites in the cis-regulatory region and the output is the rate of transcription. The simplest method of describing the relationship between the transcription factors and the output is by using boolean logic (1, 2). For example, 2 activators can regulate a gene with AND logic in which both are required or with OR logic in which either transcription factor is sufficient for transcription. Knowing which mode of combinatorial regulation a gene employs can be important for determining its function in regulatory networks. For example, the coherent feed forward loop, one of the most common motifs in gene regulatory networks (3), filters noise in upstream signals differently depending on how one of the participating genes integrates its inputs (4, 5).In general, the output of a gene will not be binary, and will depend on the concentrations of the transcription factors in a complex manner. The notion of logic operations can be generalized by introducing a continuous function that encodes the dependence of the rate of transcription on the concentrations of the inputs. Such "cis-regulatory input functions (CRIFs)" (5) have been evaluated experimentally for the well-studied lac operon. The CRIF of the wild type is complex (6, 7), but those of certain mutant operons represent molecular analogs of binary logic gates (8). In higher organisms, many genes are regulated by a large number of transcription factors, often with multiple binding sites for each factor (9-11). It is increa...

Defining cooperativity in gene regulation locally through intrinsic noise

“…6 is also the j = 1 case of Eq. 3 in our recent study on combinatorial regulation in gene networks [25]. Substituting the variance of A into the equation for the covariance, we obtain $$\langle {\xi }_A{\xi }_B\rangle =\frac{g_A{g}_B}{k_A(k_A+k_B)}true(\frac{\partial f}{\partial {\varphi }_A}true),$$ which yields a relationship between 〈 ξ A ξ B 〉 and ∂f/∂ϕ A .…”

“…We obtain these by expanding the step operator Ê and f in the master equation in powers of Ω –1/2 , as detailed elsewhere [25, 26]. The first moments (means) are trivial: $$\begin{array}{cc}right\langle N_A∕\Omega \rangle & left={\varphi }_A\\ right\langle N_B∕\Omega \rangle & left={\varphi }_B.\end{array}$$ From the deterministic rate equations (Eq.…”

“…Based on this fact, we recently suggested that higher-order correlation functions of copy numbers can provide model-free signatures of complex regulatory interactions. We specifically showed how to elucidate combinatorial contributions to regulatory interactions involving two or more transcription factors (whether they act sub- or super-additively) [25]. The approach is based on analytically derived relationships between the moments of the distribution from an expanded master equation [26] and the theoretical form of the combinatorial interaction, which is encoded in the cis -regulatory input function [27].…”

On the spontaneous stochastic dynamics of a single gene: complexity of the molecular interplay at the promoter

“…Going further by considering extrinsic stochasticity and autoregulation could be achieved with various techniques [23,34,69,70,74,75]. In the light of the highly non-linear spontaneous behavior we showed, revisiting with the present model properties of signal transmission and stochastic resonance that have been identified with simple models of promoter [22,28-30,32,38,39,43] can already be expected to reveal new properties of gene regulatory structures.…”

Fast Adaptive Uniformization of the Chemical Master Equation

“…A single equation in the CME describes the time-derivative of the probability of a certain state at all times t ≥ 0. Thus, the solution of the CME is the probability distribution over all states of the CTMC at a particular time t, that is, the transient state probabilities at time t. The solution of the CME is then used to derive measures of interest such as the distribution of switching delays [28], the distribution of the time of DNA replication initiation at different origins [32], or the distribution of gene expression products [45]. Moreover, many parameter estimation methods require the computation of the posterior distribution because means and variances do not provide enough information to calibrate parameters [21].…”