Joseph Kimmel, Richard Salter, Peter Thomas
Chemical reaction networks by which individual cells gather and process information about their chemical environments have been dubbed "signal transduction" networks. Despite this suggestive terminology, there have been few attempts to analyze chemical signaling systems with the quantitative tools of information theory. Gradient sensing in the social amoeba Dictyostelium discoideum is a well characterized signal transduction system in which a cell estimates the direction of a source of diffusing chemoattractant molecules based on the spatiotemporal sequence of ligand-receptor binding events at the cell membrane. Using Monte Carlo techniques (MCell) we construct a simulation in which a collection of individual ligand particles undergoing Brownian diffusion in a three-dimensional volume interact with receptors on the surface of a static amoeboid cell. Adapting a method for estimation of spike train entropies described by Victor (originally due to Kozachenko and Leonenko), we estimate lower bounds on the mutual information between the transmitted signal (direction of ligand source) and the received signal (spatiotemporal pattern of receptor binding/unbinding events). Hence we provide a quantitative framework for addressing the question: how much could the cell know, and when could it know it? We show that the time course of the mutual information between the cell's surface receptors and the (unknown) gradient direction is consistent with experimentally measured cellular response times. We find that the acquisition of directional information depends strongly on the time constant at which the intracellular response is filtered.
1 Introduction: gradient sensing in eukaryotes Biochemical signal transduction networks provide the computational machinery by which neurons, amoebae or other single cells sense and react to their chemical environments. The precision of this chemical sensing is limited by fluctuations inherent in reaction and diffusion processes involving a Current address: Computational Neuroscience Graduate Program, The University of Chicago. Oberlin Center for Computation and Modeling, http://occam.oberlin.edu/. To whom correspondence should be addressed. http://www.case.edu/artsci/math/thomas/thomas.html; Oberlin College Research Associate.
finite quantity of molecules [1, 2]. The theory of communication provides a framework that makes explicit the noise dependence of chemical signaling. For example, in any reaction A + B C , we may view the time varying reactant concentrations A(t) and B (t) as input signals to a noisy channel, and the product concentration C (t) as an output signal carrying information about A(t) and B (t). In the present study we show that the mutual information between the (known) state of the cell's surface receptors and the (unknown) gradient direction follows a time course consistent with experimentally measured cellular response times, reinforcing earlier claims that information theory can play a role in understanding biochemical cellular communication [3, 4]. Dictyostelium is a soil dwelling amoeba that aggregates into a multicellular form in order to survive conditions of drought or starvation. During aggregation individual amoebae perform chemotaxis, or chemically guided movement, towards sources of the signaling molecule cAMP, secreted by nearby amoebae. Quantitive studies have shown that Dictyostelium amoebae can sense shallow, static gradients of cAMP over long time scales (30 minutes), and that gradient steepness plays a crucial role in guiding cells . The chemotactic efficiency (CE), the population average of the cosine between the cell displacement directions and the true gradient direction, peaks at a cAMP concentration of 25 nanoMolar, similar to the equilibrium constant for the cAMP receptor (the Keq is the concentration of cAMP at which the receptor has a 50% chance of being bound or unbound, respectively). For smaller or larger concentrations the CE dropped rapidly. Nevertheless over long times cells were able (on average) to detect gradients as small as 2% change in [cAMP] per cell length. At an early stage of development when the pattern of chemotactic centers and spirals is still forming, individual amoebae presumably experience an inchoate barrage of weak, noisy and conflicting directional signals. When cAMP binds receptors on a cell's surface, second messengers trigger a chain of subsequent intracellular events including a rapid spatial reorganization of proteins involved in cell motility. Advances in fluorescence microscopy have revealed that the oriented subcellular response to cAMP stimulation is already well underway within two seconds [6, 7]. In order to understand the fundamental limits to communication in this cell signaling process we abstract the problem faced by a cell to that of rapidly identifying the direction of origin of a stimulus gradient superimposed on an existing mean background concentration. We model gradient sensing as an information channel in which an input signal the direction of a chemical source is noisily transmitted via a gradient of diffusing signaling molecules; and the "received signal" is the spatiotemporal pattern of binding events between cAMP and the cAMP receptors . We neglect downstream intracellular events, which cannot increase the mutual information between the state of the cell and the direction of the imposed extracellular gradient . The analysis of any signal transmission system depends on precise representation of the noise corrupting transmitted signals. We develop a Monte Carlo simulation (MCell, [10, 11]) in which a simulated cell is exposed to a cAMP distribution that evolves from a uniform background to a gradient at low (1 nMol) average concentration. The noise inherent in the communication of a diffusionmediated signal is accurately represented by this method. Our approach bridges both the transient and the steady state regimes and allows us to estimate the amount of stimulus-related information that is in principle available to the cell through its receptors as a function of time after stimulus initiation. Other efforts to address aspects of cell signaling using the conceptual tools of information theory have considered neurotransmitter release  and sensing temporal signals , but not gradient sensing in eukaryotic cells. A typical natural habitat for social amoebae such as Dictyostelium is the complex anisotropic threedimensional matrix of the forest floor. Under experimental conditions cells typically aggregate on a flat two-dimensional surface. We approach the problem of gradient sensing on a sphere, which is both harder and more natural for the ameoba, while still simple enough for us analytically and numerically. Directional data is naturally described using unit vectors in spherical coordinates, but the ameobae receive signals as binding events involving intramembrane protein complexes, so we have developed a method for projecting the ensemble of receptor bindings onto coordinates in R3 . In loose analogy with the chemotactic efficiency , we compare the projected directional estimate with the true gradient direction represented as a unit vector on S2 . Consistent with observed timing of the cell's response to cAMP stimulation, we find that the directional signal converges quickly enough for the cell to make a decision about which direction to move within the first two seconds following stimulus onset.
2 Methods 2.1 Monte Carlo simulations
Using MCell and DReAMM [10, 11] we construct a spherical cell (radius R = 7.5m, ) centered in a cubic volume (side length L = 30m). N = 980 triangular tiles partition the surface (mesh generated by DOME1 ); each contained one cell surface receptor for cAMP with binding rate k+ = 4.4 107 sec-1 M-1 , first-order cAMP unbinding rate k- = 1.1 sec-1  and Keq = k- /k+ = 25nMol cAMP. We established a baseline concentration of approximately 1nMol by releasing a cAMP bolus at time 0 inside the cube with zero-flux boundary conditions imposed on each wall. At t = 2 seconds we introduced a steady flux at the x = -L/2 wall of 1 molecule of cAMP per square micron per msec, adding signaling molecules from the left. Simultaneously, the x = +L/2 wall of the cube assumes absorbing boundary conditions. The new boundary conditions lead (at equilibrium) to a linear gradient of 2 nMol/30m, ranging from 2.0 nMol at the flux source wall to 0 nMol at the absorbing wall (see Figure 1); the concentration profile approaches this new steady state with time constant of approximately 1.25 msec. Sampling boxes centered along the planes x = 13.5m measured the local concentration, allowing us to validate the expected model behavior.
Figure 1: Gradient sensing simulations performed with MCell (a Monte Carlo simulator of cellular microphysiology, http://www.mcell.cnl.salk.edu/) and rendered with DReAMM (Design, Render, and Animate MCell Models, http://www.mcell.psc.edu/). The model cell comprised a sphere triangulated with 980 tiles with one cAMP receptor per tile. Cell radius R = 7.5m; cube side L = 30m. Left: Initial equilibrium condition, before imposition of gradient. [cAMP] 1nMol (c. 15,000 molecules in the volume outside the sphere). Right: Gradient condition after transient (c. 15,000 molecules; see Methods for details).