# Our Research | Center for Chemical Dynamics in Living Cells

**Novel Chemical Kinetics for Description of Chemical Noise in Small Heterogeneous Reaction Systems**

Our research is primarily focused on understanding biological control mechanisms in single cell biological networks, for example, the gene expression reaction network. We recently developed a novel chemical kinetics theory for the efficient description of chemical noise in general biopolymer reactions. We are currently collaborating with other theoretical physical chemistry groups on the application of a latter theory to account for the quantification of gene expression noise.

To achieve a quantitative understanding of the probabilistic behavior of living cells, we must first construct an accurate mathematical description of the intracellular networks interacting with complex cell environments. This is no easy task. To this end, we present a novel model and stochastic kinetics theory for an intracellular network that interacts with hidden cell environments. Our approach employs a complete description of cell state dynamics and its coupling to the system network. What the results reveal is that the various environmental effects on the product number fluctuation of intracellular reaction networks can be collectively characterized by a Laplace transform of the time-correlation function of the product creation rate fluctuation, with the Laplace variable being the product decay rate. Based the latter result, we propose an efficient method for the quantitative analysis of the chemical fluctuation produced by intracellular networks coupled to hidden cell environments. By applying our approach to the gene expression network, we obtain simple analytic results for gene expression variability and the environment-induced correlations between the expression levels of mutually non-interacting genes. Our theoretical results help us build a unified framework for the quantitative understanding of various gene expression statistics observed across a number of different systems that bear a small number of adjustable parameters with clear physical meanings.

Lim, Kim, Park, Yang, Song, Chang, Lee and Sung, Phys. Rev. X (2015)

Cell-to-cell variation in gene expression, also called noise, is a general phenomenon observed within cell populations. Transcription is known to be the key stage of gene expression where noise is generated, however, how variation in RNA polymerase (RNAP) concentration contributes to gene expression noise is unclear. Here, we quantitatively investigate how variations in absolute amounts of RNAP molecules affect noise in the expression of two fluorescent protein reporters driven by identical promoters. We find that intrinsic noise is independent of variation in RNAP concentrations, whereas extrinsic noise, which is variation in gene expression due to varying cellular environments, scales linearly with variation in RNAP abundance. Specifically, the propagation of RNAP abundance variation to expressed protein noise is inversely proportional to the concentration of RNAP, which suggests that the change in noise that results from RNAP fluctuations is determined by the fraction of promoters that is not occupied by RNAP.

Yang, Kim, Lim, Ki, An, Ki, Sung & Lee, Nature Comm 5, 4761

We investigate the reaction event counting statistics (RECS) of an elementary biopolymer reaction, in which the rate coefficient is dependent on states of the biopolymer and the surrounding environment, and discover a universal kinetic phase transition in the RECS of the reaction system with dynamic heterogeneity. From an exact analysis for a general model of elementary biopolymer reactions, we find that the variance in the number of reaction events is dependent on the square of the mean number of the reaction events when the size of measurement time is small on the relaxation time scale of rate coefficient fluctuations, which does not conform to renewal statistics. On the other hand, when the size of the measurement time interval is much greater than the relaxation time of rate coefficient fluctuations, the variance becomes linearly proportional to the mean reaction number in accordance with renewal statistics. Gillespie’s stochastic simulation method is generalized for the reaction system with a rate coefficient fluctuation. The simulation results confirm the correctness of the analytic results for the time dependent mean and variance of the reaction event number distribution. On the basis of the obtained results, we propose a method of quantitative analysis for the reaction event counting statistics of reaction systems with rate coefficient fluctuations, which enables one to extract information about the magnitude and the relaxation times of the fluctuating reaction rate coefficient. This methodology elminates bias that can be introduced by assuming a particular kinetic model of conformational dynamics and the conformation dependent reactivity. An exact relationship is established between a higher moment of the reaction event number distribution and the multitime correlation of the reaction rate for the reaction system with a nonequilibrium initial state distribution as well as for the system with the equilibrium initial state distribution.

Lim, Park, Park, Cao, Silbey, Sung, J. Chem. Theor. Comp. (2012)

Fluctuating turnover times of a single enzyme become observable with the advent of modern cutting-edge, single enzyme experimental techniques. Although the conventional chemical kinetics and its modern generalizations could provide a satisfactory quantitative description for the mean of the enzymatic turnover times, to the best of our knowledge there has not yet been a successful quantitative interpretation for the variance or the randomness of the enzymatic turnover times. In this review, we briefly discuss several theories in this field, and compare predictions of these theories to the randomness parameter data reported for the b-galactosidase enzyme. We find the recently proposed kinetics for renewal reaction processes could provide an excellent quantitative interpretation of the randomness parameter data. From the analysis of the randomness parameter data of the single enzyme reaction, one can extract quantitative information about the mean lifetime of the enzyme-substrate complex, the success or the failure probability of the catalytic reaction per each formation of ES complex, and the non-Poisson character of the reaction dynamics of the ES complex (which is beyond reach of the long-standing paradigm of the conventional chemical kinetics theory).