Modeling and Simulation of Bacterial Self-Organization inCircular Container Along Contact Line as Detected by Bioluminescence Imaging

Mathematical modeling and numerical simulation of quasi-one dimensional spatiotemporal pattern formation along the three phase contact line in the fluid cultures of lux gene engineeredEscherichia coli is investigated in this paper. The numerical simulation is based on a one-dimensional-inspace mathematical model of a bacterial self-organizationas detected by quasi-one-dimensional bioluminescence imagi n . The pattern formation in a luminous E. coli colony was mathematically modeled by the nonlinear reaction-diffusi onchemotaxis equations. The numerical simulation was carrie d out using the finite difference technique. Regular oscillat ions as well as chaotic fluctuations similar to the experimental o nes were computationally simulated. The influence of the signal dependent as well as density-dependent chemotactic sensit ivity, the non-local sampling and the diffusion nonlinearity on the pattern formation was investigated. The computational simulations showed that a constant chemotactic sensitivity, a l ocal sampling and a linear diffusion can be applied for modeling the formation of the bioluminescence patterns in a colony of luminous E. coli. Keywords-chemotaxis; reaction-diffusion; pattern formation; simulation; whole-cell biosensor . I. I NTRODUCTION This paper is an extension of work originally reported in The Third International Conference on Advances in System Simulation [1]. Various microorganisms respond to certain chemicals found in their environment by migrating towards higher (chemoattraction) or lower (chemorepulsion) concentrati ons of the substance. The directed movement of microorganisms in response to chemical gradients is called chemotaxis [2]. Chemotaxis plays a crucial role in a wide range of biological phenomena, e.g., within the embryo, the chemotaxis affects avian gastrulation and patterning of the nervous system [3] . Often, microorganisms not only move up chemical gradients towards a chemoattractant, but they are also able to produce more of the chemoattractant. This is the effect that produce s the aggregation of the motile microorganisms into local clusters with high density and hence results in a pattern formation [4]. Although the chemotaxis has been observed in many bacterial species,Escherichia coli is one of the mostly studied examples. E. coli responds to the chemical stimulus by alternating the rotational direction of their flagella [2 ], [3]. Various mathematical models based on the Patlak-KellerSegel model have been successfully used as important tools to study the mechanisms of the chemotaxis [5]. An excellent review on the mathematical modeling of the chemotaxis has been presented by Hillen and Painter [6]. Bacterial species including E. coli have been observed to form various patterns under different environmental condi tions [4], [7], [8]. Bacterial populations are capable of se lforganization into states exhibiting strong inhomogeneiti es in density [9], [10]. Recently, the spatiotemporal pattern s in the fluid cultures ofE. coli have been observed by employing lux-gene engineered cells and a bioluminescence imaging technique [11], [12]. However, the mechanisms governing the formation of bioluminescence patterns still remain unclear. Over the last two decades, lux-gene engineered bacteria have been successfully used to develop whole cell-based biosensors [13]. A whole-cell biosensor is an analyte probe consisting of a biological element, such as a genetically en gineered bacteria, integrated with an electronic componen t to yield a measurable signal [14]. Whole-cell biosensors have been successfully used for the detection of environmental pollutant bioavailability, various stressors, including dioxins, endocrine-disrupting chemicals, and ionizing radiation [ 15]. To solve the problems currently limiting the practical use o f whole-cell biosensors, the bacterial self-organization w ithin the biosensors have to be comprehensively investigated. In this paper, the bacterial self-organization in a small circular container near the three phase contact line is investigated [11], [12]. A computational model for efficient simulating the formation of the spatiotemporal patterns experimentally detected by quasi-one-dimensional biolum inescence imaging in the fluid cultures of E. coli has recently been developed [16], [17]. The pattern formation in a luminousE. coli colony was modeled by the nonlinear reaction-diffusion-chemotaxis equations. The mathemati cal model was formulated in a one-dimensional space. Several 155 International Journal on Advances in Systems and Measurements, vol 5 no 3 & 4, year 2012, http://www.iariajournals. org/systems_and_measurements/ 2012, © Copyright by authors, Published under agreement with IARIA www.iaria. org different model variations were analyzed, and a minimal model was obtained for simulating the formation of the bioluminescence patterns representing the self-organiza tion of luminousE. coli. The aim of this work was to improve the already existing computational model by introducing the nonlinear diffusio n of cells, the non-local sampling and several kinds of the chemotactic sensitivity [6]. By extending the model in this way, the improvements of the patterns simulated using extended model were expected. In this paper, the pattern formation is computationally investigated assuming two kinds of the chemotactic sensitivity, the signal-dependen t sensitivity and the density-dependent sensitivity. The no nlocal sampling and the nonlinear diffusion are investigate d individually and collectively. The numerical simulation a t transient conditions was carried out using the finite difference technique [18]. The computational model was validated by experimental data. Regular oscillations as well as chaotic fluctuations similar to experimental ones were computationally simulated. By varying the input parameter s the output results were analyzed with a special emphasis on the influence of the model parameters on the spatiotemporal pattern formation in the luminous E. coli colony. The rest of the paper is organized as follows. Section II provides a state of the art on the mathematical modeling of bacterial self-organization. Section III describ es the mathematical model of the bacterial self-organization in a circular container. The computational modeling of a physical experiment is discussed in Section IV. Section V is devoted to the results of the numerical simulation where the effects of different chemotactic sensitivity function s, the non-local gradient and the diffusion nonlinearity are investigated. Finally, the main conclusions are summarize d in Section VI. II. M ODELS OFBACTERIAL SELF-ORGANIZATION Different mathematical models based on advectionreaction-diffusion equations have already been developed for computational modeling the pattern formation in bacterial colonies [7], [8], [19], [20], [21]. The system of coupled nonlinear partial differential equations introduced by Ke ller and Segel are still among the most widely used [5], [6]. According to the Keller and Segel approach, the main biological processes can be described by a system of two conservation equations ( x ∈ Ω, t > 0), ∂n ∂t = ∇ (Dn∇n− h(n, c)n∇c) + f(n, c), ∂c ∂t = ∇ (Dc∇c) + gp(n, c)n− gd(n, c)c, (1) wherex and t stand for space and time, n(x, t) is the cell density,c(x, t) is the chemoattractant concentration, Dn(n) andDc are the diffusion coefficients, f(n, c) stands for cell growth and death, h(n, c) stands for the chemotactic sensitivity, gp andgd describe the production and degradation of the chemoattractant [5], [21]. Both diffusion coefficients,Dn andDc, are usually assumed to be constant. However, the nonlinear cell diffusion depending on the chemoattractant concentration or/and the cell density is also considered [6]. In this work, we conside r the nonlinear diffusion of the form