Optimal Strategy in E. coli Chemotaxis: An Information Theoretic Approach Lin Wang and Sima Setayeshgar Department of Physics, Indiana University, Bloomington, Indiana Physical constants for motion: Cell speed: μm/sec Mean run time: 1 sec Mean tumble time: 0.1 sec The chemotaxis signal transduction pathway in E. coli is one of the best-characterized chemotaxis network, all of the genes and proteins involved in its chemotaxis network are known and most of them have been crystallized. Body size:1 μ m in length, 0.4 μ m in radius Flagellum:10 μ m long, 45 nm in diameter Motivation Biochemical signaling is the most fundamental level of information processing in biological systems, where an external stimulus is measured and converted into a response. E. coli Chemotaxis Chemotaxis, a cell’s motion toward desirable chemicals (usually nutrients) and away from harmful ones, is achieved through alternating ‘runs’ and ‘tumbles’. The mean run-time is modulated in response to the cells’s measurement of the chemoattractant concentration, resulting in a biased random walk up (down) chemoattractant (repellant) concentration gradients. Adaptation Adaptation is an important and generic property of biological systems. Adaptive responses occur over a wide range of time scales, from fractions of a second in neural systems, to millions of years in the evolution of species. In bacterial chemotaxis, adaptation occurs when the response (e.g., running bias) returns precisely to the pre- stimulus level while the stimulus persists. It allows the system to compensate for the presence of continued stimulation and to be ready to respond to further stimuli. Numerical Implementation The chemotaxis signal transduction pathway in E. coli – a network of ~50 interacting proteins – converts an external stimulus (change in concentration of chemo- attractant / repellent) into an internal stimulus (change in concentration of intracellular response regulator, CheY-P) which in turn interacts with the flagella motor to bias the cell’s motion. Signal Transduction Pathway Motor Response [CheY-P] Stimulus Flagellar Bundling Motion Photon counting in vision [1,2] We use the well-characterized chemotaxis network in E. coli as a prototype for exploring general principles governing information processing in biological signaling networks. [1] R. C. Hardie et al. (2001) Nature 413, [2] M. Postma et al. (1999) Biophysical Journal [3] S. M. Block et al Cell Chemotaxis network Adaptation [4] Attractant: 30 μ M aspartate. Repellent: 100 μ M NiCl 2 Adaptation to various step change of aspartate. Blue: 1 μM; Red: 100 μM. (simulation) nP 1 (n)P 2 (n) MoleculeNumberConcentration (μM) Y Yp00 R E6276- B Bp00 Chemotaxis Network Equations and Parameters Table I: Signal Transduction Network Table III: Initial Protein Levels Table II: Activation Probabilities Motor response A simple threshold model [6] is used to model motor response. The motor switches state whenever CheY-P trace (blue trace) crosses the threshold (red line). Simulating Reactions Reactions are simulated using Stochsim [5] package, a general platform for simulating reactions stochastically. Symbols: n: Number of molecules in reaction system n 0 : Number of pseudo-molecules N A : Avogadro constant p: Probability for a reaction to happen Δt: Simulation time step V: Simulation volume Bi-molecular reaction Uni-molecular reaction Focus E. coli varies its response to input signals with different statistics. Our goal is to understand how signal transduction pathways, such as the chemotaxis network, may adapt to the statistics of the fluctuating input so as to optimize the cell’s response. We construct a measure of the information transmission rate and investigate the role of varying response. [5] C. J. Morton-Firth et al J. Theor. Biol [6]T. Emonet et al Bioinformatics Mutual Information The average information that observation of Y provides about the signal X, is I, the mutual information of X and Y [7]. I is at minimum, zero, when Y is independent of X, while it is at maximum when Y is completely determined by X. The I/O mutual information rate can be calculated by the following equation [8]. s: input signal; P(s): probability distribution of signal r: response; P(r): probability distribution of response r(s): I-O relation, mapping s to r. n: noise P(n|r): probability distribution of noise distribution conditioned on response [7] Spikes, Fred Rieke et al. 1997, p [8] N. Brenner et al. (2000) Neuron Adaptation variation Molecule counting in chemotaxis [3] Photon Δ[Ca 2+ ] Δ[Na + ] et al. Attractant Δ[CheY-P] Response of drosophila photoreceptor to single photon absorption. Response of E. coli to external attractant. From R. M. Berry, Encyclopedia of Life Sciences Fluorescently labeled E. coli (Berg lab) Model Validation By utilizing this realistic and stochastic numerical implementation, we explore E. coli chemotaxis network from the standpoint of general information-processing concepts. Input-Output Relation Adaptation [9] Motor CCW and CW intervals [11] Adaptation time [10] Discussion: the simulation results are in good agreement with experiments, although the adaptation times differ by a small factor. Cell response when exposed to a step change of aspartate from 0 to 0.1 mM (left), 10 μ M (right) beginning at 5 sec. Transition time to step change of external attractant. Distribution of wild-type E. coli motor CW (grey) and CCW (black) intervals. Upper: Gaussian distributed signal ( μ =3 μ M, σ 2 = μ, τ = 1 sec) Lower panel: Response to the input signal. I/O relation under signals with different statistics. ( τ = 1 sec) The response is the average of responses in each bin of signal. [9] S. M. Block et al Cell [10] H. C. Berg et al PNAS [11] T. Emonet et al Bioinformatics E. coli chemotaxis network Signal Output Input signal Artificially generated Gaussian distributed time series with correlation time τ. Output Number of CheY-P molecules Simulation Experiment The chemotaxis network is able to extract as much as information possible once the input signal varies slower relative to the response time of the chemotaxis network. Under an input signal with specific statistics, the chemotaxis network varies its response to optimize the cell’s performance, maximizing the mutual information between input signal and output response. Conclusions Effect of Correlation Time τ My first step is to investigate the effect of correlation time τ to the I/O mutual information rate of the chemotaxis network. Response r(s) to signals: μ =1 μ M, σ 2 = μ, τ = 0.1, 0.3, 0.8, 1 sec, respectively. At τ > 0.8 sec, the response does not change any more. (This holds true for signals with different mean values) Effect of τ in I/O mutual information The I/O mutual information rate of E. coli chemotaxis network is plotted as a function of correlation time τ. The Gaussian distributed signals used here have means of 1, 3, 5, and 10, respectively. Use a realistic description of motor to replace the simple threshold model of motor response. Take into account the clustering effect among trans-membrane aspartate receptors to improve the performance of the numerical implementation. Investigate role of adaptation time. Future Work Effect of τ on I/O relation Effect of varying response Use r (s 1 ) under input signal s 1 (μ 1 =1 μM, σ 1 2 = μ 1, τ 1 = 1 sec) to find P(r) for different input signals, and calculate the mutual information between r (s 1 ) and s k. The calculated I/O mutual information rate of E. coli chemotaxis network maximizes under the condition that the response and the input signal matches. Input to our system (E. coli chemotaxis network) is the concentration of attractant, and the output is the number of CheY-P molecules. I thank Sima Seteyashgar for the help in preparing this poster, and thank Xianfeng for useful suggestions. Acknowledgment [4] Sourjik et al. (2002) PNAS