From indication to decision: A hierarchical approach to model the chemotactic behavior of Escherichia coli

Highlights

• We provided a new Stochastic Multi-Layer model to simulating bacterial motility.

• We reduced the chemotaxis to its underlying physical and chemical processes.

• We proposed a non-homogeneous Markovian random walk in modeling of bacterial chemotaxis.

• We used stochastic simulation in chemical layer and drag force of flagella at low Reynolds numbers.

Abstract

Reducing the complex behavior of living entities to its underlying physical and chemical processes is a formidable task in biology. Complex behaviors can be characterized as decision making: the ability to process the incoming information via an intracellular network and act upon this information to choose appropriate strategies. Motility is one such behavior that has been the focus many modeling efforts in the past. Our aim is to reduce the chemotactic behavior in Escherichia coli to its molecular constituents in order to paint a comprehensive and end-to-end picture of this intricate behavior. We utilize a hierarchical approach, consisting of three layers, to achieve this goal: at the first level, chemical reactions involved in chemotaxis are simulated. In the second level, the chemical reactions give rise to the mechanical movement of six independent flagella. At the last layer, the two lower layers are combined to allow a digital bacterium to receive information from its environment and swim through it with verve. Our results are in concert with the experimental studies concerning the motility of E.coli cells. In addition, we show that our detailed model of chemotaxis is reducible to a non-homogeneous Markov process.

Introduction

Decision making is defined as choosing a course of action from a set of possibilities (Kitajima and Toyota, 2013). Biological systems have to cope with both internal and external perturbations and make the “right” decisions amidst this pandemonium. The decision-making machinery is shaped by natural selection to fit the conditions of its environment (Tagkopoulos, Liu, Tavazoie, 2008, Mitchell, Romano, Groisman, Yona, Dekel, Kupiec, Dahan, Pilpel, 2009). Motility can be viewed as a decision-making process that benefits the living cell by enabling it to find resources in its niche more efficiently (Xie and Wu, 2014). Cell motility requires sensors to monitor the environment, actuators to act upon the incoming information, and an network to process that information.

The majority of bacteria are motile, swimming being its most common form Jarrell and McBride (2008) and Lauga (2016). Early studies revealed a substantial amount of variation in motility of clonal cells as they navigate a uniform environment (Dufour et al., 2016). In a homogeneous environment with uniformly-distributed resources, all decisions apropos of motility would be equally likely to be taken -i.e., random walk. Consequently in such circumstances, a motile cell would randomly navigate the environment. Having encountered a non-uniform distribution of resources in the environment, a motile cell will move to more resource-rich areas - i.e., the default random walk turns into biased random walk.

Chemotaxis, the ability of bacterial cells to sense chemical cues in their environment and move accordingly, predates the divergence of the eubacteria from the archaebacteria (Woese and Fox, 1977). The steps taken in chemotactic behavior, to seek attractants and avoid repellents, can be seen as a chain of biased random steps. To illustrate this point, we can focus on Escherichia coli. E. coli detects the concentration of chemoattractants in its vicinity via an array of sensors, processes the sensory data via a sensory network, and swims accordingly using its flagella (Sourjik, Wingreen, 2012, Frankel, Pontius, Dufour, Long, Hernandez-Nunez, Emonet, 2014). Following a trail of chemoattractants to get to their source is seemingly an insurmountable obstacle for E. coli, since their small size means that the difference between the amount of chemoattractants around its head and its tail would not be meaningful, and, consequently, useless in finding the correct direction. In reality, by rotating its flagella clockwise (CW) or counter clockwise (CCW), E. coli runs and tumbles through the environment (Wadhams, Armitage, 2004, Shimizu, Tu, Berg, 2010).

Many mathematical models have been developed to understand the bacterial chemotaxis (reviewed in Tindall et al., 2008b). The early models focused on the adaptive behavior of individual bacteria in different environmental conditions at a macroscopic level (Segel, 1976, Spudich, Koshland Jr, 1976, Block, Segall, Berg, 1982, Block, Segall, Berg, 1983). Some models (e.g., Goldbeter and Koshland Jr, 1982), used ordinary differential equations to describe the bacterial response to a gradient of chemical stimulants. Some used the Ising model (Shi, Duke, 1998, Duke, Bray, 1999, Shi, 2000, Shi, 2001, Shi, 2002, Guo, Levine, 1999, Guo, Levine, 2000), others utilized an individual-based approach (Frankel, Pontius, Dufour, Long, Hernandez-Nunez, Emonet, 2014, Niu, Wang, Duan, Li, 2013), and some emphasized the hydrodynamic aspects of swimming (Elgeti et al., 2015) and the role of drift versus diffusion (Chatterjee et al., 2011). However some researches focused on collective motility patterns (Elgeti, Winkler, Gompper, 2015, Colin, Drescher, Sourjik, 2019, Tindall, Maini, Porter, Armitage, 2008)

Most theoretical models of chemotaxis are limited to incorporating a single motor or simply assume that all cells have a single flagellum (Bray, Levin, Lipkow, 2007, Kalinin, Jiang, Tu, Wu, 2009, Matthäus, Jagodič, Dobnikar, 2009, Jiang, Ouyang, Tu, 2010, Flores, Shimizu, ten Wolde, Tostevin, 2012, Kanehl, Ishikawa, 2014). Constructing a comprehensive model of bacterial chemotaxis from the single-flagellum state has remained out of reach (Mears et al., 2014). What is more, most models do not explain how macroscopic chemotaxis behavior arises from the fundamental laws of chemistry and physics. In this paper, we propose a model that reduce chemotaxis to simple phenomena.

In our Stochastic Multi-Layer (SML) model, E. coli is treated like a minute biological submarine. This nano-submarine is propelled by an average of six flagella in low Reynolds’ number regime. Our model attempts to offer a comprehensive description of chemotactic behavior in E. coli by breaking this complex process into three levels. In the first level, chemoattractants react with the receptors, causing molecular events in the cell that can result in the rotation of each flagellum. The sensory network determines the direction and rotation rate of each flagellum. In the second level, each flagellum generates a force and the resultant force of all flagella causes the E. coli to move in the direction of this force. In the third level, the combination of different force vectors of each flagellum provides a range of direction and length of movement in each step - i.e., the behavior of the bacterium emerges from the chemical and the physical levels. At each step, as the concentration of chemoattractants sensed by the bacteria changes, so does the distribution of probability of all choices, i.e., the direction and the distance of travel at that step.

Here, we attempt to present a simple non-homogeneous Markov model, alongside an equivalent hierarchical model, to illustrate how a biased random walk be a successful strategy to seek nutrients in an environment. The hierarchical model is designed to include biochemical processes of chemotaxis within individual cells and the associated motion of cells within a 2D environment. This model presents an end-to-end picture of chemotaxis and with a simplified version of the underpinning molecular mechanisms. In our view, while to details of our three-level model reflects the intricacies of the biology, its behaviour would be indistinguishable from a non-homogeneous Markovian random walk (NHMRW) process (as a negative control). Given the similarity between the NHMRW and the hierarchical model. the NHMRW can be used to decrease the computational complexity and cost of simulating chemotaxis and allows the markedly more efficient simulation of bacterial chemotaxis in a large population).