Spatio-temporal Pattern Formation due to Host-Circuit Interplay in Gene Expression Dynamics

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Spatio-temporal Pattern Formation due to Host-Circuit

Interplay in Gene Expression Dynamics

Priya Chakrabortya, Mohit K. Jollyb, Ushasi Roy1b, Sayantari Ghosh2a

aDepartment of Physics, National Institute of Technology, Durgapur, 713209, West

Bengal, India

bCentre for BioSystems Science and Engineering, Indian Institute of

Science, Bangalore, 560012, Karnataka, India

Abstract

Biological systems are majorly dependent on their property of bistability in

order to exhibit nongenetic heterogeneity in terms of cellular morphology and

physiology. Spatial patterns of phenotypically heterogeneous cells, arising

due to underlying bistability, may play signiﬁcant role in phenomena like

bioﬁlm development, adaptation, cell motility etc. While nonlinear positive

feedback regulation, like cooperative heterodimer formation are the usual

reason behind bistability, similar dynamics can also occur as a consequence

of host-circuit interaction. In this paper, we have investigated the pattern

formation by a motif with non-cooperative positive feedback, that imposes

a metabolic burden on its host due to its expression. In a cellular array

set inside diﬀusible environment, we investigate spatio-temporal diﬀusion in

one dimension as well as in two dimension in the context of various initial

conditions respectively. Moreover, the number of cells exhibiting the same

steady state, as well as their spatial distribution has been quantiﬁed in terms

of connected component analysis. The eﬀect of diﬀusion coeﬃcient variation

has been studied in terms of stability of related states and time evolution of

patterns.

Keywords: Pattern formation, Reaction diﬀusion system, Non-cooperative

Gene regulation, Emergent bistability.

1Corresponsing author: ushasiroy@iisc.ac.in

2Corresponsing author: sayantari.ghosh@phy.nitdgp.ac.in

arXiv:2210.06099v1 [q-bio.QM] 12 Oct 2022

1. Introduction

Proteins are responsible for diverse functionalities, serving the cells for

structural support, motility, enzymatic activity, inner organization, interac-

tion with the outside environment and many more [1]. From DNA to mRNA,

and then to proteins, the information ﬂows in a tightly controlled way in-

side cell. This ﬂow of information, i.e., gene expression has two major steps:

transcription and translation, taking places in cell nucleus and cytoplasm

respectively. Transcriptional gene regulation is one of the fundamental ways

that control expression of any particular gene in terms of location, amount

and timing. Though all the cells in a isogenic microbial population contains

same genome, switching the expression of a gene ON and OFF, the cell pop-

ulation can get bifurcated into two subpopulations, which are distinct, but

coexisting. Bistability, a vastly preferred physical behavior of living cells,

is known to regulate this “nongenetic”, phenotypic heterogeneity [2,3,4].

In bistable response, protein concentration attains any of the two drastically

diﬀerent steady states (low or high response), and shows a history-dependent

behaviour, hysteresis. The phenomena introduces a memory in the system,

making it retain its state instead of variation or ﬂuctuations in inducer level.

The essential nonlinearity required for bistability is conventionally achieved

by a genetic system through a positive feedback with cooperative regulation

by multimer formation, or by a combination of more than one feedback loops.

However, recently, bistability driven by host-circuit coupling have been de-

picted by researchers [5,6]. In [5], the authors termed this phenomena as

emergent bistability, where, along with a positive non-cooperative gene reg-

ulation, a secondary double-negative feedback loop provides the necessary

nonlinearity, indirectly originating due to host-circuit coupling. While tox-

icity of expressed protein may be a reason behind growth retardation of

host cell [7], this phenomena can be much prevalent in natural systems also

[8,9,10]. A possible explanation lies in the fact that in presence of limited

resources, the protein synthesis may impose a metabolic burden, causing a

reduction in the amount of resources available for cellular growth.

The connection between spatially and/or temporally structured phenotypic

heterogeneity with diversiﬁcation and adaptation of populations have been

explored in several works [11,12,13]. Though, in most of the related works,

environmental ﬂuctuations and variations are associated with this patterning,

in his seminal work, A. Turing had already shown that two interacting dif-

fusing chemicals can generate a stable inhomogeneous pattern, under certain

conditions [14]. Connecting this biological systems, mathematical models

have been formed to explore the dynamics of pattern formation in Activator-

Inhibitor system [15,16,17,18], feedback quenched oscillator system [19] and

many more [20]. Eventually successful experimental demonstrations have

provided a strong background to these theoretical models [21,22,23]. How-

ever, the lesser explored area of pattern formation in gene regulatory systems

has drawn the attention of research community very recently and diﬀerent

mechanisms are employed to study realistic scenarios [24,25,26,27,28].

In this work, we explore the eﬀect of the inherent stochasticity due to spa-

tial diﬀusion, in presence of host-circuit coupling to observe spatially and

temporally structured pattern formation. For this purpose we consider the

emergent bistable dynamics we have mentioned before [5]. Though, for this

dynamics, successful implementation with synthetic gene circuits have been

achieved and stochastic responses have also been studied [5,29], no signiﬁ-

cant study on the pattern formation in a diﬀusive environment by the system

is done as per our knowledge. Moreover, in all these studies a single, isolated

genetic circuit has been considered along with its host cell. On the other

hand, a collection of host cells, each with the circuit of interest embedded,

evolving in presence of diﬀusion of the synthesized protein, is a scenario more

closely relatable with experiments. So, we investigate the spatio-temporal be-

haviour in this thorough study. Here, in this paper we described the model

formulation in Sec. 2, deterministic analyses related to equilibria and bifur-

cation is described in Sec. 3and the detailed results of reaction diﬀusion

system is explained in Sec. 4. Further quantitative analysis of this host-

circuit interaction driven spatio-temporal pattern formation is done in Sec.

5. Finally, in Sec. 6, we concluded with brief discussion on relevance of the

present work and future perspective.

2. Model formulation

Let us describe the model formulation of the concerned motif as shown

in Fig. 1. Let Ube a protein which activates its own synthesis, with an

eﬀective synthesis rate constant, α. We consider no coopertivity associated

in this positive feedback. Basal synthesis rate of the protein Uis repre-

sented by δ. Growth causing the increase in volume dilutes the protein, we

denote this dilution rate can achieve the maximum value φ. Now, to incor-

porate host-circuit interaction, we consider that the expression of protein U

is associated with an expense of resources present in cell. This eﬀectively

Figure 1: Schematic diagram of model motif which shows emergent bistability. Protein U

activates its own synthesis by a positive feedback loop (thick blue arrow) and has a natural

decay rate (black arrow). Synthesis of the protein puts a metabolic burden aﬀecting cellular

growth. As growth is hindered, protein dilution gets reduced creating a double negative

feedback loop (marked by consecutive hammerheads). This acts eﬀectively as a positive

feedback.

creates a metabolic burden on cell growth aﬀecting the protein dilution. We

consider βand γto be the linear and nonlinear reduction in dilution rate

due to this host-circuit coupling. The natural degradation rate has been

taken into account by ∆U. The idea of considering nonlinear degradation as

a consequence of metabolic burden has been explored in other works [5,6]

and brieﬂy explained by Monod in [30]. Another valid explanation includes

the possibilities of direct/indirect toxic eﬀect of protein synthesis on growth

of the cell [7]. Now, the form of mathematical representation of the above

considerations are given by Eq. 1.

dt =δ+α U

1 + U−φ U

β+γ U −∆UU(1)

We proceed with further analysis with this model equation.

3. Results: Deterministic System

3.1. Equilibria & Stability:

Biologically signiﬁcant equilibria for this given system should be non-

negative solutions where du

dt = 0. In other words, if we express Eq. 1in

Figure 2: Bifurcation analysis of the model motif. (a) System equation f(U) vs. Uplot.

f(U) intersects the Uaxis in 3 diﬀerent points, representing 3 steady states of the system

in phase space. Among these 2 are stable point represented in solid green circle and one

is unstable point represented in red hollow dot. Black arrows indicates the direction of

ﬂow lines in phase space. (b) Growth function, f1(U) and decay function, f2(U) is plotted

against U. Two curve intersects in 3 diﬀerent points, generating 3 possible solution of the

system. (c) Protein Ushows bistability wrt. γ. Parameter values are φ= 10, α= 3 for the

red curve and α= 5 for the blue curve. (d) Protein Ushows bistability wrt. α. Parameter

values are φ= 20, γ = 2.5 for red curve, γ= 10 for the blue curve. (e) Bistability curve

of protein Uwrt. φ. Parameter values are α= 5, γ= 10 for blue curve, γ= 5 for the red

curve. (f) Bifurcation diagram in the plane of parameter α, the synthesis rate of protein

vs. parameter φ, maximum dilution rate of protein. Bistable region is shown in blue-gray

color with a deﬁned blue boundary which separates it from two monostable region. For

(a), (b), (f) γ= 10. For (a), (b) α= 8.5, φ = 20. For all curves δ= 0.1, β = 1.15,∆U= 1.

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摘要：

Spatio-temporalPatternFormationduetoHost-CircuitInterplayinGeneExpressionDynamicsPriyaChakrabortya,MohitK.Jollyb,UshasiRoy1b,SayantariGhosh2aaDepartmentofPhysics,NationalInstituteofTechnology,Durgapur,713209,WestBengal,IndiabCentreforBioSystemsScienceandEngineering,IndianInstituteofScience,Bangalore...

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Spatio-temporal Pattern Formation due to Host-Circuit Interplay in Gene Expression Dynamics

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