In-silico Implementation of Bacterial Chemotaxis Lin Wang Advisor: Sima Setayeshgar
Chemotaxis in E. coli Dimensions: Body size: 1 μm in length 0.4 μm in radius Flagellum: 10 μm long Physical constants: Cell speed: μm/sec Mean run time: 1 sec Mean tumble time: 0.1 sec From Berg Lab From R. M. Berry, Encyclopedia of Life Sciences
From Single Cells to Populations … Chemotactic response of individual cells forms the basis of macroscopic pattern formation in populations of bacteria: Colonies Pattern formation in E. coli: From H.C. Berg and E. O. Budrene, Nature (1995) Biofilms Agrobacterium biofilm: From Fuqua Lab
Motivation Chemotaxis as a well-characterized “ model ” signaling network, amenable to quantitative analysis and extension to other signaling networks from the standpoint of general information-processing concepts, such as signal to noise, adaptation and memory Chemotaxis as an important biophysical mechanism, for example underlying initial stages of biofilm formation
Modeling Chemotaxis in E. coli Signal Transduction Pathway Motor Response [CheY-P] Stimulus Flagellar Response Motion
Outline Chemotaxis signal transduction network in E. coli Stochastic implementation of reaction network using Stochsim Flagellar and motor response Preliminary numerical results
Chemotaxis Signal Transduction Pathway in E. coli
Ligand Binding E: receptor complex a: ligand (eg., aspartate) Rapid equilibrium: Rates 1 : E: K D = 1.71x10 -6 M -1 E * : K D = 12x10 -6 M -1 [1] Morton-Firth et al., J. Mol. Biol. (1999)
Receptor Activation E n : methylated receptor complex; activation probability, P 1 (n) E n a: ligand-bound receptor complex; activation probability, P 2 (n) E n * : active form of E n E n * a: active form of E n a Table 1: Activation Probabilities nP 1 (n)P 2 (n)
Methylation R: CheR E n (a): E n, E n a E n (*) (a): E n, E n *, E n a, E n * a Rate constants: k 1f = 5x10 6 M -1 sec -1 k 1r = 1 sec -1 k 2f = sec -1 (1) (2)
Demethylation Bp: CheB-P E n * (a): E n *, E n * a Rate constants: k 1f = 1x10 6 M -1 sec -1 k 1r = 1.25 sec -1 k 2f = sec -1 (1) (2)
Autophosphorylation E * :E n *, E n * a Rate constant: k f = 15.5 sec -1
CheY Reactions Y: CheY Yp: CheY-P Rate constants: k 1f = 1.24x10 -3 sec -1 k 1r = 4.5x10 -2 sec -1 k 2f = sec -1
CheY Phosphotransfer Rate constants: k 1f = 5x10 6 M -1 sec -1 k 2f = 20 sec -1 k 2r = 5x10 6 M -1 sec -1 k 3f = 7.5 sec -1 k 3r = 5x10 6 M -1 sec -1
CheB Reactions B: CheB Bp: CheB-P Rate constant: k f = 0.35 sec -1
CheB Phosphotransfer Rate constants: k 1f = 5x10 6 M -1 sec -1 k 2f = 16 sec -1 k 2r = 5x10 6 M -1 sec -1 k 3f = 16 sec -1 k 3r = 5x10 6 M -1 sec -1
Simulating Reactions Stochastic 2 : Reaction has probability P of occurring a) Generate x, a uniform random number in [0, 1]. b) x <= P, reaction happens. c) x > P, reaction does not happen. How to generate P from reaction rates? [2] Morton-Firth et al., J. Mol. Biol. (1998) Two methods: Deterministic: ODE description, using rate constants,
Stochsim Package Stochsim package is a general platform for simulating reactions using a stochastic method.
Pseudo-molecule Pseudo-molecules are used to simulate unimolecular reaction. Number of pseudomolecule in simulating system: k 1max : fastest unimolecular reaction rate k 2max : fastest bimolecular reaction rate
From Rate Constant to Probability Unimolecular reaction n: number of molecules from reaction system n 0 : number of pseudomolecules N A : Avogadro constant Bimolecular reaction
Simulation Parameters Reaction Volume: 1.41 x liter Rate constants given above. Table 2: Initial Numbers of Molecules MoleculeNumberConcentration (μM) Y Yp00 R E4246- B Bp00
Output of Signal Transduction Network Fig 1. Number of CheY-P molecules as a function of time, the trace is smoothed by an averaging window of 0.3 sec. The motor switches state whenever threshold (red line) is crossed. It ’ s assumed that there is only 1 motor/cell.
Flagellar Response Flagellar state directly reflects motor state, except that 20% of the motor changing from CCW to CW is dropped 3. Assume there is only 1 flagellum/cell. [3] Alon et al., The EMBO Journal (1998)
Motion Motion of the cell is determined by the state of flagellum. CCW run CW tumble
Run and Tumble Process Run 4 Tumble 5 t t+Δt α v = 20 μm/s D r = s -1 γ = 4 μ = -4.6 β = [4] Zou et al., Biophys. J. (2003)[5] Berg and Brown, Nature (1972)
Some Simulation Results Distribution of run and tumble intervals. Diffusion of a population of cells in an unbounded region in the absence of stimulus. Diffusion of a population of cells in a bounded region (z>0), with and without stimulus.
Motor CW and CCW Intervals Fig 2. Fraction of motor CW/CCW intervals of wild-type cell in an environment without ligand. Left: Experiment (Korobkova et al., Nature 2004); Right: Simulation
Diffusion in Unbounded Region: No Stimulus Fig 3. Mean-squared distance from initial position as a function of time (averaged over 540 cells). Diffusion constant is found to be 4.4 * mm 2 /s, consistent with experimental results 6. [6] Paul Lewus et al., BioTech. and BioEng. (2001)
Diffusion in Bounded Region (z>0) Fig 4. Number of cells (out of a total of 540) above z=1.2 mm as a function of time. Red: constant linear gradient of aspartate zM/μM; Blue: no aspartate.
Future Directions Optimal biochemical signal processing (role of “ adaptive ” network adaptation time) Role of chemotaxis in initial stages of biofilm formation Realistic description of chemotaxis in E. coli to explore:
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