Spatial Heterogeneity in Bacterial Cells ⋆ Carlos Barajas * Domitilla Del Vecchio * * Department of Mechanical Engineering, MIT, Cambridge, MA 02139, USA,(e-mail: carlobar@mit.edu, ddv@mit.edu) Abstract: Commonly used models of genetic circuits assume a well-mixed ensemble of species. However, experimental data suggests that appreciable spatial heterogeneity exists in bacteria cells. There exists no unified modeling framework to capture this spatial phenomena. To this end, we model spatial heterogeneity inside bacterial cells and propose a simple framework that accounts for spatial information. In this document, we start with a generic spatial-temporal partial differential equitation (PDE) model. Then, we exploit time scale separation between diffusion and the reaction dynamics to derive a reduced model consisting solely of ordinary differential equations (ODEs). This result is then applied to study an enzymatic-like reaction. It is shown that spatial heterogeneity modifies the binding strength between two species that reversibly bind to each other. We show that the modified binding rate for certain cases can be larger or smaller than that of a well-mixed model. Therefore, this work takes a step forward towards creating a general and simple framework to model spatial heterogeneity in bacterial cells and thus improving the predictive power of current models that are used to design genetic circuits Keywords: genetic circuits, model reduction, biomolecular systems, reaction-diffusion 1. INTRODUCTION Deterministic models of gene networks typically assume a well-mixed ensemble of species inside the cell (Del Vecchio and Murray (2017)). However, it is well known that spa- tial heterogeneity is prevalent inside the cell (Wingreen and Huang (2015); Weng and Xiao (2014)). Depending on the origin of replication, plasmids tend to localize within bacterial cells (Wang et al. (2016)). Furthermore, chromosome genes are distributed in the cell according to the chromosomes complex spatial structure. In bacterial cells, any species freely diffusing through the chromosome (e.g., mRNA, ribosome, and protease) experiences what are known as excluded volume effects, which is the ten- dency for the species to be ejected from the nucleoid due to the space occupied by the dense DNA mesh (Castellana et al. (2016)). These excluded volume effects for ribosomes and RNAP in bacteria have been observed experimentally (Bakshi et al. (2012)). Despite the strong evidence against a well-mixed model, no standard modeling framework exists for genetic circuits that captures the spatial-temporal organization inside the cell. Furthermore, current approaches that rely solely on numerical simulations of partial differential equations (PDEs) may be impractical for genetic circuit design. In this abstract, we present a model reduction strategy starting from a system of coupled PDEs and ODEs to a reduced system of ODEs via timescale separation be- tween diffusion and the reaction dynamics. By applying this result to an enzymatic-like reaction, we demonstrate that the reduced model accounts for spatial heterogene- ity in the spaced averaged dynamics by multiplying the ⋆ This work was supported by NSF award number 1521925 association rate constant of bimolecular interactions by a correction factor that depends on spatial information. Thus, this reduced model has similar computational cost as current well-mixed models, yet it captures spatial ef- fects. Specifically, we focus on capturing excluded vol- ume effects and gene location information. We analyze the correction factor in two different cases: when the en- zyme and substrate both diffuse (mRNA-sRNA, protein- protease- mRNA-ribosomes) and when the enzyme diffuses and the substrate is fixed in space (transcription factor- DNA,protein-membrane). This analysis provides insight into how effectively species interact depending on their size and the location where they are fixed. 2. RESULTS Notation : Let z =[z 1 ,...,z n ] T ∈ R n (where superscript T denotes the transpose operation) and the j -th component of z is denoted by z j . A vector of zeros is denoted as 0 n = [0,..., 0] T ∈ R n and we use A = diag(u) ∈ R n×n to refer to a square matrix with all zeros in the off-diagonals and diagonal elements specified by the vector u ∈ R n . R n + denotes the positive orthant of R n . Let Ω = (0, 1), Ω = [0, 1], ∂ Ω= {0, 1}. The model used to capture intracellular species interacting as they diffuse inside the cell is now introduced. As in Castellana et al. (2016), we assume the cell to have a cylindrical geometry, angular symmetry, and radial ho- mogeneity such that the concentration of a species varies only axially (the spatial x direction). Symmetry relative to the mid-cell is assumed and hence only half of the cell is considered; x ∈ [0, 1], where x = 0 is at the mid-cell and x = 1 is at the cell poles. Furthermore, we assume a constant cross-sectional area along the axial Preprints of the 21st IFAC World Congress (Virtual) Berlin, Germany, July 12-17, 2020 Copyright lies with the authors 16957