In-silico Implementation of Bacterial Chemotaxis
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In-silico Implementation of Bacterial ChemotaxisLin WangAdvisor: Sima Setayeshgar
Chemotaxis in E. coliDimensions: Body size: 1 m in length 0.4 m in radius Flagellum: 10 m long
Physical constants: Cell speed: 20-30 m/secMean run time: 1 secMean tumble time: 0.1 sec
From Berg LabFrom 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:ColoniesPattern formation in E. coli:From H.C. Berg and E. O. Budrene, Nature (1995)BiofilmsAgrobacterium biofilm:From Fuqua Lab
MotivationChemotaxis 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. coliSignal TransductionPathwayMotor Response[CheY-P]StimulusFlagellar ResponseMotion
OutlineChemotaxis 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 BindingE: receptor complexa: ligand (eg., aspartate)
E: KD = 1.71x10-6 M-1E*: KD = 12x10-6 M-1 Morton-Firth et al., J. Mol. Biol. (1999)
Receptor ActivationEn: methylated receptor complex; activation probability, P1(n)Ena: ligand-bound receptor complex; activation probability, P2(n)En*: active form of En En*a: active form of Ena
Table 1: Activation Probabilities
MethylationR: CheREn(a): En, EnaEn(*)(a): En, En*, Ena, En*a
Rate constants:k1f = 5x106 M-1sec-1k1r = 1 sec-1k2f = 0.819 sec-1
DemethylationBp: CheB-PEn*(a): En*, En*a
Rate constants:k1f = 1x106 M-1sec-1k1r = 1.25 sec-1k2f = 0.15484 sec-1
Rate constant:kf = 15.5 sec-1
CheY ReactionsY: CheYYp: CheY-P
Rate constants:k1f = 1.24x10-3 sec-1k1r = 4.5x10-2 sec-1k2f = 14.15 sec-1
CheY PhosphotransferRate constants:k1f = 5x106 M-1sec-1k2f = 20 sec-1k2r = 5x106 M-1sec-1k3f = 7.5 sec-1k3r = 5x106 M-1sec-1
CheB ReactionsB: CheBBp: CheB-P
Rate constant:kf = 0.35 sec-1
CheB PhosphotransferRate constants:k1f = 5x106 M-1sec-1k2f = 16 sec-1k2r = 5x106 M-1sec-1k3f = 16 sec-1k3r = 5x106 M-1sec-1
Simulating ReactionsStochastic2: Reaction has probability P of occurringa) Generate x, a uniform random number in [0, 1].b) x P, reaction does not happen.
How to generate P from reaction rates?
 Morton-Firth et al., J. Mol. Biol. (1998)Two methods:Deterministic: ODE description, using rate constants,
Stochsim package is a general platform for simulating reactions using a stochastic method.
Pseudo-moleculePseudo-molecules are used to simulate unimolecular reaction.
Number of pseudomolecule in simulating system:
k1max: fastest unimolecular reaction ratek2max: fastest bimolecular reaction rate
From Rate Constant to ProbabilityUnimolecular reaction n: number of molecules from reaction system n0: number of pseudomolecules NA: Avogadro constant
Simulation Parameters Reaction Volume: 1.41 x 10-15 liter Rate constants given above.Table 2: Initial Numbers of Molecules
Output of Signal Transduction NetworkFig 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. Its assumed that there is only 1 motor/cell.
Flagellar state directly reflects motor state, except that 20% of the motor changing from CCW to CW is dropped3.
Assume there is only 1 flagellum/cell.
 Alon et al., The EMBO Journal (1998)
Motion of the cell is determined by the state of flagellum.
CCW runCW tumble
Run and Tumble ProcessRun4
Tumble5v = 20 m/sDr = 0.06205 s-1 = 4 = -4.6 = 18.32 Zou et al., Biophys. J. (2003) Berg and Brown, Nature (1972)
Some Simulation ResultsDistribution 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 IntervalsFig 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 StimulusFig 3. Mean-squared distance from initial position as a function of time (averaged over 540 cells). Diffusion constant is found to be 4.4 * 10-4 mm2/s, consistent with experimental results6. 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 10-8 zM/M; Blue: no aspartate.
Future DirectionsOptimal biochemical signal processing (role of adaptive network adaptation time)
Role of chemotaxis in initial stages of biofilm formationRealistic description of chemotaxis in E. coli to explore:
First, design simple reactions show how this worked.
Second, find all reactions and compare the configuration files to those used in Agentcell package. Specifically, I will talk about modeling chemotaxis in E. coli. As is familiar to this audience, E. coli is the work horse of molecular biology. We focus on it from a modeling standpoint because it is well-studied. In particular, the chemotaxis network in E. coli is the best characterized two-component signaling system: all protein components are known and most have been crystallized.
As you know, chemotaxis is the cells biased random walk toward attractants and away from repellents. It consists of runs (approximately 50 microns in length) punctuated by tumbles.
Each cell consists of 4-6 flagella. Flagella have an inherent helicity, such that CCW rotation of the flagellar motor leads to bundling of the flagella, which then act as a propeller leading to running motion, and CW rotation of one or more of the flagellar motors leads to the flying apart of the bundle and tumbling motion.
The cells response to its environment is through modulation of its mean run time (through modulation of the CW/CCW motor bias), extending it in favorable directions and suppressing it in unfavorable directions.
The chemotactic response of individual cells is important in macroscopic pattern formation in populations of bacteriafrom colonies shown on the left, where it is believed that only chemotaxis and depletion of the agar substrate are responsible for the observed complex patternstobiofilms shown on the right, where it is believed that chemotaxis is important in the initial, monolayer phase of biofilm formationOur group is interested in chemotaxis from two standpoints:
- as a well-characterized example of intracellular signaling networks, 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
- as an important biophysical mechanism, for example underlying the initial stages of biofilm formation.In modeling chemotaxis in E. coli, we must describe the signal transduction pathway that converts an external stimulus (attractant or repellent) into an internal response regulator (change in concentration of CheY-P), as well as the motor and flagellar response leading to the cells motion.
E. colis chemotaxis network is a multiscale network. Some reactions are fast (ligand binding, phosphrylation, and phosphotransfer) while there are other slow reactions (methylation and demethylation), allowing for a memory of the past environment. Adaptation is achieved by balancing the these fast and slow processes.The outline of this talk is as follows:
I will give a quick review of the details of the chemotaxis signal transduction network in E. coli.
I will describe a stochastic implementation of this reaction network using a package called Stochsim developed by Dennis Brays group at Cambridge.
I will describe our modeling of the flagellar and motor response, based on AgentCell developed by University of Chicago and Argonne National Lab groups.
Finally, I will give some preliminary results demonstrating our successful numerical implementation.Where are these KD from?
Rapid equilibrium: If a MS is chosen, stochsim will first equilibrate any rapid equilibria defined for this complex.
Rate constants1:kf = 1x109 M-1sec-1kr = 1x103 sec-1
. Carl Jason et al. 1998
Mention: In the subsequent slides, the rates are based on the same paper.
Rates: Biemann & Koshland, 1994 Aspartate receptors of E. coli and Salmonella typhimurium bind ligand with negative and half-of-the sites cooperativity. Biochem. 33, 629-634
In viro In vitro
Barkai: 1.0e09 M-1sec-1 1.0e03 sec-1Spiro: 7e07 M-1sec-1 70sec-1Possible question:
Katherine A. Borkovich, Lisa A. Alex, and Melvin I. Simon, Attenuation of sensory receptor signaling by covalent modification, Proc. Natl. Acad. Sci. USA, 89(1992) 6756Katherine A. Borkovich, et al. Transmembrand Signal transduction in bacterial chemotaxis involves ligand-depdent activation of phosphate group transfer. Proc. Natl. Acad. Sci. USA, 86(1989) 1208
Determined by in vitro experiment for Tar receptor , 0:2:4 are determined and 1, 3 methyl groups are calculated by linear interpolation.
Ligand-bound Tar receptor, in vitro experiment.
Barkai: unoccupied: 0.1 0.5 0.72 1 occupied 0 0.1 0.5 1
Following adaptation to high level aspartate, the average methylation level rises from two to three.
For unoccupide Tar with 3 methyl groups, the probability is calculated
deltaG = -RTln(p/(1-p))This reaction is in question
En should be inactive, since they assume R only binds to inactive form of En.
Have problem with this slide.
Barkai: 8e07 M-1sec-1100 sec-1 0.1sec-1Spiro: k1r/k1f = 1.7 e-06 M here 0.2 e-06 M use different rates for receptors with different methyl groups.Also have question of the second reaction. Does EnBp changes to inactive form before it can be dephosphrylated. Or both active and inactive form.
Barkai: 8e08 M-1sec-11000sec-1 0.1sec-1Spiro: dose not given. Diffusion limited reaction rate:
Personal communication.The second reaction is Yp desphosphorylation by CheZ.
SpiroTotal : 3 e07 M-1sec-1Second not considered
The first one is assumed to be diffusion-limited reaction.
Association are also assumed to be diffusion-limited. But 5 times 1e06.
Dissociation constant od E-YP is 4 uM. (Li et al. 1995)Dissociation constant of E-Y is 1.5 uM. (In vitro experiment. stewart)
What is E in the first reaction: All kind of forms of E. En(*)(a)What is E in the second reaction.
In vitro experiment.
3 e 05 M-1s-1
Dissociation constant of EB is 3.2 uM. (Li et al.
Dissociation constant of EpB is assumed to be the same as EB.Association constant are assumed to be diffusion-limited.Why do we use a stochastic method?Its more realistic, since reactions are taken place between individual proteins.At low concentration, rate constant does not work well.
LIN: add reference for Stochsim here.Number of pserdomolecules is calculated to minimise the stiffness between the unimolecular and bimolecular reactions. The probability of the fastest unimolecular and fastest bimolecular reactions are as close as possible.
Why do we need to minimize the stiffnesss? (????)
In the system, we want as much unimolecular reaction to occur as bimolecular reaction. Thats why we need to balance the probability of these two kinds of reactions.
Stiff: For a multiscale network, how do we simulate it. It we use a small time step interval, we can not get a good simulation of the slow reaction. If we use a large time step interval, the information of the fast reaction will lost.
Comparison with Gilepse algorithm?
Gilespe randomly choose a reaction each step, determine the time step and whether this reaction will happen. So it does not need a concept of pseudo-molecule to simulate a unimolecular reaction. But its not efficient in simulating multi-state receptor complex.Threshold crossing
1 motor only.Why is there a 20% drop rate?
1 motor only.
Why do we use these parameters, in other word, where do they come from?
P(delta(Theta)) = 1 / sqrt(2*pi*sigma) * exp(-x^2 / (2*sigma^2)); sigma = 2 * D_r * tRun:
P(a) fits the distribution of tumbling angles measure by Berg and Brown (1972)
Mean value or shape?The mean value of this distribution is 68.6801And the variance is 37.
Run and Tumble intervals? OR
Motor CW and CCW intervals? (In Agentcell paper, they use motor CW and CCW interval.)Diffusion:
4.45 x 10^(-4) mm^2 / sec