Finding stationary solutions to the chemical master equation by gluing state spaces at one or two states recursively

Noise is indispensible to key cellular activities, including gene expression coordination and probabilistic differentiation. Stochastic models, such as the chemical master equation (CME), are essential to model noise in the levels of cellular components. In the CME framework, each state is associated with the molecular counts of all component species, and specifies the probability for the system to have that set of molecular counts. Analytic solutions to the CME are rarely known but can bring exciting benefits. For instance, simulations of biochemical reaction networks that are multiscale in time can be sped up tremendously by incorporating analytic solutions of the slow time-scale dynamics. Analytic solutions also enable the design of stationary distributions with properties such as the modality of the distribution, the mean expression level, and the level of noise. One way to derive the analytic steady state response of a biochemical reaction network was recently proposed by (Mélykúti et al. 2014). The paper recursively glues simple state spaces together, for which we have analytic solutions, at one or two states. In this work, we explore the benefits and limitations of the gluing technique proposed by Mélykúti et al., and introduce recursive algorithms that use the technique to solve for the analytic steady state response of stochastic biochemical reaction networks. We give formal characterizations of the set of reaction networks whose state spaces can be obtained by carrying out single-point gluing of paths, cycles or both sequentially. We find that the dimension of the state space of a reaction network equals the maximum number of linearly independent reactions in the system. We then characterize the complete set of stochastic biochemical reaction networks that have elementary reactions and two-dimensional state spaces. As an example, we propose a recursive algorithm that uses the gluing technique to solve for the steady state response of a mass-conserving system with two connected monomolecular reversible reactions. Even though the gluing technique can only construct finite state spaces, we find that, by taking the size of a finite state space to infinity, the steady state response can converge to the analytic solution on the resulting infinite state space. Finally, we illustrate the aforementioned ideas with the example of two interconnected transcriptional components, which was first studied by (Ghaemi and Del Vecchio 2012)

A Bayesian approach to inferring chemical signal timing and amplitude in a temporal logic gate using the cell population distributional response

Stochastic gene expression poses an important challenge for engineering robust behaviors in a heterogeneous cell population. Cells address this challenge by operating on distributions of cellular responses generated by noisy processes. Similarly, a previously published temporal logic gate considers the distribution of responses across a cell population under chemical inducer pulsing events. The design uses a system of two integrases to engineer an E. coli strain with four DNA states that records the temporal order of two chemical signal events. The heterogeneous cell population response was used to infer the timing and duration of the two chemical signals for a small set of events. Here we use the temporal logic gate system to address the problem of extracting information about chemical signal events. We use the heterogeneous cell population response to infer whether any event has occurred or not and also to infer its properties such as timing and amplitude. Bayesian inference provides a natural framework to answer our questions about chemical signal occurrence, timing, and amplitude. We develop a probabilistic model that incorporates uncertainty in the how well our model captures the cell population and in how well a sample of measured cells represents the entire population. Using our probabilistic model and cell population measurements taken every five minutes on generated data, we ask how likely it was to observe the data for parameter values that describe square-shaped inducer pulses. We compare the likelihood functions associated with the probabilistic models for the event with the chemical signal pulses turned on versus turned off. Hence, we can determine whether an event of chemical induction of integrase expression has occurred or not. Using Markov Chain Monte Carlo, we sample the posterior distribution of chemical pulse parameters to identify likely pulses that produce the data measurements. We implement this method and obtain accurate results for detecting chemical inducer pulse timing, length, and amplitude. We can detect and identify chemical inducer pulses as short as half an hour, as well as all pulse amplitudes that fall under biologically relevant conditions

A Bayesian approach to inferring chemical signal timing and amplitude in a temporal logic gate using the cell population distributional response

Stochastic gene expression poses an important challenge for engineering robust behaviors in a heterogeneous cell population. Cells address this challenge by operating on distributions of cellular responses generated by noisy processes. Similarly, a previously published temporal logic gate considers the distribution of responses across a cell population under chemical inducer pulsing events. The design uses a system of two integrases to engineer an E. coli strain with four DNA states that records the temporal order of two chemical signal events. The heterogeneous cell population response was used to infer the timing and duration of the two chemical signals for a small set of events. Here we use the temporal logic gate system to address the problem of extracting information about chemical signal events. We use the heterogeneous cell population response to infer whether any event has occurred or not and also to infer its properties such as timing and amplitude. Bayesian inference provides a natural framework to answer our questions about chemical signal occurrence, timing, and amplitude. We develop a probabilistic model that incorporates uncertainty in the how well our model captures the cell population and in how well a sample of measured cells represents the entire population. Using our probabilistic model and cell population measurements taken every five minutes on generated data, we ask how likely it was to observe the data for parameter values that describe square-shaped inducer pulses. We compare the likelihood functions associated with the probabilistic models for the event with the chemical signal pulses turned on versus turned off. Hence, we can determine whether an event of chemical induction of integrase expression has occurred or not. Using Markov Chain Monte Carlo, we sample the posterior distribution of chemical pulse parameters to identify likely pulses that produce the data measurements. We implement this method and obtain accurate results for detecting chemical inducer pulse timing, length, and amplitude. We can detect and identify chemical inducer pulses as short as half an hour, as well as all pulse amplitudes that fall under biologically relevant conditions