1. Optimal Strategy in E. coli Chemotaxis: An Information Theoretic Approach Lin Wang and Sima Setayeshgar Department of Physics, Indiana University, Bloomington, Indiana 47405 Physical constants of motion: Cell speed: 20-30 μm/sec Mean run time: 1 sec Mean tumble time: 0.1 sec E. coli is the best-studied organism in molecular biology, providing an experimentally accessible basis for our understanding of fundamental cellular processes. 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 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. Numerical Implementation The chemotaxis signal transduction pathway in E. coli is a network of interacting proteins that 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 mean runtime. It is a model system for studying the properties of the two-component superfamily of receptor-regulated phosphorylation pathways in general. 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, 186-193 [2] M. Postma et al. (1999) Biophysical Journal 77 1811-1823 [3] S. M. Block et al. 1982 Cell 31 215-226 Perfect Adaptation [4] Attractant: 30 μ M aspartate. Repellent: 100 μ M NiCl 2 Adaptation to various step changes of aspartate. Blue: 1 μM; Red: 100 μM (simulation) nP 1 (n)P 2 (n) 00.020.00291 10.170.02 20.50.17 30.8740.5 40.9970.98 MoleculeNumberConcentration (μM) Y1568418 Yp00 R2500.29 E6276- B19282.27 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 in maximizing this rate. [5] C. J. Morton-Firth et al. 1998 J. Theor. Biol.. 192 117-128 [6]T. Emonet et al. 2005 Bioinformatics 21 2714-2721 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 minimum (zero) when Y is independent of X, while it is maximum when Y is completely determined by X. The Input-Output (I/O) mutual information rate, I, is given by: 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, p122-123 [8] N. Brenner et al. (2000) Neuron. 26 695-702 Adaptation Time Molecule counting in chemotaxis [3] Photon Δ[Ca 2+ ], Δ[Na + ], etc. Attractant Δ[CheY-P] Response of drosophila photoreceptor to photon absorption Response of E. coli to change in external attractant concentration From R. M. Berry, Encyclopedia of Life Sciences Courtesy of H. C. Berg lab) Model Validation Utilizing this realistic numerical implementation, we explore the chemotaxis network in E. coli from the standpoint of information-processing: Input-Output Relation Adaptation [9] Motor CCW and CW intervals [11] Adaptation time [10] Discussion: Our simulation results are in good agreement with experiments, providing an experimentally faithful computational framework for bacterial chemotaxis. Cell response (probability of CCW rotation of flagella, leading to running motion) when exposed to a step change of aspartate from 0 to 0.1 mM (left), 10 μ M (right) beginning at 5 sec Variable network adaptation time in response to different step changes of concentration of external attractant, demonstrating increase in adaptation time with increasing stimulus step size Distribution of CW (grey) and CCW (black) intervals in wild-type adapted cells Upper: Input = Gaussian distributed signal as a function of time {μ=3 μM, σ 2 = μ, τ = 1 sec}. Lower : Output = system response to the input signal, as a function of time. Response, r(s i ), to input signals with {μ i, σ i 2, τ = 1 sec} Response function, r(s), as a function of input signal. Because of the stochastic nature, response to input signal varies. Each point represents the average value of response. [9] S. M. Block et al. 1982 Cell 31 215-226 [10] H. C. Berg et al. 1975 PNAS 72 3235-3239 [11] T. Emonet et al. 2005 Bioinformatics 21 2714-2721 E. coli chemotaxis network Signal Output Input signal Gaussian distributed time series for chemoattractant concentration with correlation time,  : Output Number of CheY-P molecules Simulation Experiment The chemotaxis network in E. coli functions under varying environmental conditions. We have shown that as the statistics of the input stimulus change, the input-output relation varies. This adaptive behavior allows E. coli to extract “as much information as possible” from the input signal (by maximizing the mutual information between the input and output). Conclusions Varying Statistics of Input Response r(s) to signals with μ=1 μM, σ 2 = 1 μM 2, τ = 0.1, 0.3, 0.8, 1 sec, respectively. For τ > 1 sec, the response does not vary significantly with . (This also holds true for signals with different mean values). Effect of τ on 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 μ=1, 3, 5, and 10 μM, respectively. Work in progress includes investigation of: 1)Motor bias as the output of the chemotaxis network, by constructing a more physically realistic description of the motor response based on the statistical mechanics of switching between CW/CCW states. 2)Role of the variable network adaptation time, from the standpoint of optimizing information transmission. Future Work Effect of τ on I/O relation Effect of varying response The response, r (s 1 ), to input signal s 1 (with μ 1 =1 μM, σ 1 2 = μ 1,  1 = 1 sec) is used to map different input signals s k to output r’ k (instead of using correct response r(s k ) to each s k ). The mutual information between r’ k and s k is calculated as: The I/O mutual information rate is maximized when the response and the input signal are matched. [4] Sourjik et al. (2002) PNAS. 99 123-127 Finding the input-output relation, r(s) Here,input, s : chemoattractant concentration output, r : CheY-P concentration Chemotaxis in E. coli - motion toward desirable chemicals and away from harmful ones - is an important behavioral response also shared by many other prokaryotic and eukaryotic cells. It is achieved through a series of modulated ‘runs’ and ‘tumbles’, leading to a biased random walk in the desired direction. Discussion: Intuitively, more information can be transmitted when input signal changes slowly. We show that as the time scale of changes in the input signal becomes comparable to the E. coli impulse response time (  >0.8sec), the information transmission rate approaches a constant asymptotic value. As the statistics of the input stimulus varies, E. coli’s response adapts so as to maximize the mutual information between the input signal and the output. In bacterial chemotaxis, adaptation occurs when the steady state response (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. Furthermore, the dynamical properties of the network, such as the adaptation time, vary for different inputs.