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135 nips-2006-Modelling transcriptional regulation using Gaussian Processes

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Author: Neil D. Lawrence, Guido Sanguinetti, Magnus Rattray

Abstract: Modelling the dynamics of transcriptional processes in the cell requires the knowledge of a number of key biological quantities. While some of them are relatively easy to measure, such as mRNA decay rates and mRNA abundance levels, it is still very hard to measure the active concentration levels of the transcription factor proteins that drive the process and the sensitivity of target genes to these concentrations. In this paper we show how these quantities for a given transcription factor can be inferred from gene expression levels of a set of known target genes. We treat the protein concentration as a latent function with a Gaussian process prior, and include the sensitivities, mRNA decay rates and baseline expression levels as hyperparameters. We apply this procedure to a human leukemia dataset, focusing on the tumour repressor p53 and obtaining results in good accordance with recent biological studies.

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Modelling transcriptional regulation using Gaussian processes Neil D. [sent-1, score-0.154]

2 uk Abstract Modelling the dynamics of transcriptional processes in the cell requires the knowledge of a number of key biological quantities. [sent-17, score-0.207]

3 While some of them are relatively easy to measure, such as mRNA decay rates and mRNA abundance levels, it is still very hard to measure the active concentration levels of the transcription factor proteins that drive the process and the sensitivity of target genes to these concentrations. [sent-18, score-0.977]

4 In this paper we show how these quantities for a given transcription factor can be inferred from gene expression levels of a set of known target genes. [sent-19, score-0.678]

5 We treat the protein concentration as a latent function with a Gaussian process prior, and include the sensitivities, mRNA decay rates and baseline expression levels as hyperparameters. [sent-20, score-0.562]

6 We apply this procedure to a human leukemia dataset, focusing on the tumour repressor p53 and obtaining results in good accordance with recent biological studies. [sent-21, score-0.114]

7 Microarray technology now allows measurement of mRNA abundance on a genomewide scale, and techniques such as chromatin immunoprecipitation (ChIP) have largely unveiled the wiring of the cellular transcriptional regulatory network, identifying which genes are bound by which transcription factors. [sent-23, score-0.761]

8 mRNA decay rates), most of them are very hard to measure with current techniques, and have therefore to be inferred from the available data. [sent-27, score-0.115]

9 One can formulate a large scale simpliﬁed model of regulation (for example assuming a linear response to protein concentrations) and then combine network architecture data and gene expression data to infer transcription factors’ protein concentrations on a genome-wide scale. [sent-29, score-1.027]

10 Alternatively, one can formulate a realistic model of a small subnetwork where few transcription factors regulate a small number of established target genes, trying to include the ﬁner points of the dynamics of transcriptional regulation. [sent-31, score-0.513]

11 In this paper we follow the second approach, focussing on the simplest subnetwork consisting of one tran- scription factor regulating its target genes, but using a detailed model of the interaction dynamics to infer the transcription factor concentrations and the gene speciﬁc constants. [sent-32, score-0.737]

12 In these studies, parametric models were developed describing the rate of production of certain genes as a function of the concentration of transcription factor protein at some speciﬁed time points. [sent-36, score-0.864]

13 Markov chain Monte Carlo (MCMC) methods were then used to carry out Bayesian inference of the protein concentrations, requiring substantial computational resources and limiting the inference to the discrete time-points where the data was collected. [sent-37, score-0.234]

14 We show here how a Gaussian process model provides a simple and computationally efﬁcient method for Bayesian inference of continuous transcription factor concentration proﬁles and associated model parameters. [sent-38, score-0.6]

15 Firstly, it allows for the inference of continuous quantities (concentration proﬁles) without discretization, therefore accounting naturally for the temporal structure of the data. [sent-41, score-0.062]

16 Secondly, it avoids the use of cumbersome interpolation techniques to estimate mRNA production rates from mRNA abundance data, and it allows us to deal naturally with the noise inherent in the measurements. [sent-42, score-0.202]

17 Finally, it greatly outstrips MCMC techniques in terms of computational efﬁciency, which we expect to be crucial in future extensions to more complex (and realistic) regulatory networks. [sent-43, score-0.094]

18 The paper is organised as follows: in the ﬁrst section we discuss linear response models. [sent-44, score-0.091]

19 These are simpliﬁed models in which the mRNA production rate depends linearly on the transcription factor protein concentration. [sent-45, score-0.629]

20 Although the linear assumption is not veriﬁed in practice, it has the advantage of giving rise to an exactly tractable inference problem. [sent-46, score-0.054]

21 We then discuss how to extend the formalism to model cases where the dependence of mRNA production rate on transcription factor protein concentration is not linear, and propose a MAP-Laplace approach to carry out Bayesian inference. [sent-47, score-0.777]

22 1 Linear Response Model Let the data set under consideration consist of T measurements of the mRNA abundance of N genes. [sent-51, score-0.121]

23 We consider a linear differential equation that relates a given gene j’s expression level xj (t) at time t to the concentration of the regulating transcription factor protein f (t), dxj = Bj + Sj f (t) − Dj xj (t) . [sent-52, score-1.088]

24 dt (1) Here, Bj is the basal transcription rate of gene j, Sj is the sensitivity of gene j to the transcription factor and Dj is the decay rate of the mRNA. [sent-53, score-1.059]

25 Crucially, the dependence of the mRNA transcription rate on the protein concentration (response) is linear. [sent-54, score-0.675]

26 Assuming a linear response is a crude simpliﬁcation, but it can still lead to interesting results in certain modelling situations. [sent-55, score-0.132]

27 [1] to model a simple network consisting of the tumour suppressor transcription factor p53 and ﬁve of its target genes. [sent-57, score-0.522]

28 The equation given in (1) can be solved to recover xj (t) = Bj + kj exp (−Dj t) + Sj exp (−Dj t) Dj t f (u) exp (Dj u) du (2) 0 where kj arises from the initial conditions, and is zero if we assume an initial baseline expression level xj (0) = Bj /Dj . [sent-59, score-0.532]

29 We will model the protein concentration f as a latent function drawn from a Gaussian process prior distribution. [sent-60, score-0.439]

30 This implies immediately that the mRNA abundance levels will also be modelled as a Gaussian process, and the covariance function of the marginal distribution p (x1 , . [sent-62, score-0.229]

31 , xN ) can be worked out explicitly from the covariance function of the latent function f . [sent-65, score-0.113]

32 If the covariance function associated with f (t) is given by kf f (t, t ) then elementary functional analysis yields that cov (Lj [f ] (t) , Lk [f ] (t )) = Lj ⊗ Lk [kf f ] (t, t ) . [sent-67, score-0.204]

33 Explicitly, this is given by the following formula t kxj xk (t, t ) = Sj Sk exp (−Dj t − Dk t ) t exp (Dj u) exp (Dk u ) kf f (u, u ) du du. [sent-68, score-0.452]

34 0 (4) 0 If the process prior over f (t) is taken to be a squared exponential kernel, 2 kf f (t, t ) = exp − (t − t ) l2 , where l controls the width of the basis functions1 , the integrals in equation (4) can be computed analytically. [sent-69, score-0.399]

35 The resulting covariances are obtained as √ πl [hkj (t , t) + hjk (t, t )] (5) kxj xk (t, t ) = Sj Sk 2 where 2 hkj (t , t) = exp (γk ) Dj + Dk −exp [− (Dk t + Dj )] erf Here erf(x) = x 0 2 √ π t −t − γk l exp [−Dk (t − t)] erf t − γk l exp −y 2 dy and γk = + erf (γk ) Dk l 2 . [sent-70, score-0.631]

36 To infer the protein concentration levels, one also needs the “cross-covariance” terms between xj (t) and f (t ), which is obtained as t kxj f (t, t ) = Sj exp (−Dj t) exp (Dj u) kf f (u, t ) du. [sent-74, score-0.749]

37 (6) 0 Again, this can be obtained explicitly for squared exponential priors on the latent function f as √ πlSj t −t t 2 exp (γj ) exp [−Dj (t − t)] erf − γj + erf + γj . [sent-75, score-0.503]

38 kxj f (t , t) = 2 l l Standard Gaussian process regression techniques [see e. [sent-76, score-0.098]

39 Alternatively, they can be assigned vague gamma prior distributions and estimated a posteriori using MCMC sampling. [sent-81, score-0.069]

40 In practice, we will allow the mRNA abundance of each gene at each time point to be corrupted by some noise, so that we can model the observations at times ti for i = 1, . [sent-82, score-0.453]

41 , T as, yj (ti ) = xj (ti ) + j (ti ) (8) 2 0, σji with j (ti ) ∼ N . [sent-85, score-0.153]

42 The covariance of the noisy process is simply obtained as 2 2 2 2 Kyy = Σ + Kxx , with Σ = diag σ11 , . [sent-87, score-0.087]

43 Also, all the quantities in equation (1) are positive, but one cannot constrain samples from a Gaussian process to be positive. [sent-98, score-0.087]

44 Modelling the response of the transcription rate to protein concentration using a positive nonlinear function is an elegant way to enforce this constraint. [sent-99, score-0.766]

45 In this case the induced distribution of xj (t) is no longer a Gaussian process. [sent-102, score-0.082]

46 The functional derivative of the log-likelihood with respect to f is then obtained as T N δ log p(Y |f ) (xj (ti ) − yj (ti )) =− Θ(ti − t) g (f (t))e−Dj (ti −t) 2 δf (t) σji i=1 j=1 (11) where Θ(x) is the Heaviside step function and we have omitted the model parameters for brevity. [sent-105, score-0.106]

47 In these and the following formulae ti is understood to mean the index of the grid point corresponding to the ith data point, whereas tp and tq correspond to the grid points themselves. [sent-115, score-0.595]

48 We can then compute the gradient and Hessian of the (discretised) un-normalised log posterior Ψ(f ) = log p(Y |f ) + log p(f ) [see 8, chapter 3] Ψ(f ) = log p(Y |f ) − K −1 f (15) Ψ(f ) = −(W + K −1 ) where K is the prior covariance matrix evaluated at the grid points. [sent-116, score-0.276]

49 The Laplace approximation to the log-marginal likelihood is then (ignoring terms that do not involve model parameters) ˆ ˆ ˆ (16) log p(Y ) log p(Y |f ) − 1 f T K −1 f − 1 log |I + KW |. [sent-118, score-0.105]

50 The gradient of the log-marginal with respect to the kernel parameters is [8] ˆ ∂ log p(Y |Ξ) ∂ fp ∂ log p(Y |Ξ) ∂K −1 ˆ 1 ∂K ˆ = 1 f T K −1 K f − 2 tr (I + KW )−1 W + (17) 2 ˆ ∂Ξ ∂Ξ ∂Ξ ∂Ξ ∂ fp p ˆ where the ﬁnal term is due to the implicit dependence of f on Ξ. [sent-120, score-0.324]

51 3 Example: exponential response As an example, we consider the case in which g (f (t) , θj ) = Sj exp (f (t)) (18) which provides a useful way of constraining the protein concentration to be positive. [sent-122, score-0.538]

52 Substituting equation (18) in equations (13) and (14) one obtains T N ∂ log p(Y |f ) (xj (ti ) − yj (ti )) = −∆ Θ (ti − tp ) Sj efp −Dj (ti −tp ) 2 ∂fp σji i=1 j=1 T Wpq = −δpq N ∂ log p(Y |f ) −2 2 + ∆2 Θ (ti − tp ) Θ (ti − tq ) σji Sj efp +fq −Dj (2ti −tp −tq ) . [sent-123, score-0.674]

53 ∂fp i=1 j=1 The terms required in equation (17) are, ∂ log p(Y |Ξ) 1 = −(AW )pp − ˆ 2 ∂ fp where A = (W + K −1 −1 ) . [sent-124, score-0.19]

54 Aqq Wqp q ˆ ∂f ∂K = AK −1 ∂Ξ ∂Ξ ˆ log p(Y |f ) , 3 Results To test the efﬁcacy of our method, we used a recently published biological data set which was studied using a linear response model by Barenco et al. [sent-125, score-0.192]

55 This study focused on the tumour suppressor protein p53. [sent-127, score-0.269]

56 mRNA abundance was measured at regular intervals in three independent human cell lines using Affymetrix U133A oligonucleotide microarrays. [sent-128, score-0.156]

57 The authors then restricted their interest to ﬁve known target genes of p53: DDB2, p21, SESN1/hPA26, BIK and TNFRSF10b. [sent-129, score-0.119]

58 They estimated the mRNA production rates by using quadratic interpolation between any three consecutive time points. [sent-130, score-0.081]

59 They then discretised the model and used MCMC sampling (assuming a log-normal noise model) to obtain estimates of the model parameters Bj , Sj , Dj and f (t). [sent-131, score-0.087]

60 To make the model identiﬁable, the value of the mRNA decay of one of the target genes, p21, was measured experimentally. [sent-132, score-0.09]

61 Also, the scale of the sensitivities was ﬁxed by choosing p21’s sensitivity to be equal to one, and f (0) was constrained to be zero. [sent-133, score-0.064]

62 Their predictions were then validated by doing explicit protein concentration measurements and growing mutant cell lines where the p53 gene had been knocked out. [sent-134, score-0.444]

63 1 Linear response analysis We ﬁrst analysed the data using the simple linear response model used by Barenco et al. [sent-136, score-0.218]

64 Raw data was processed using the mmgMOS model of [4], which also provides estimates of the credibility associated with each measurement. [sent-138, score-0.123]

65 Data from the different cell lines were treated as independent instantiations of f but sharing the model parameters {Bj , Sj , Dj , Ξ}. [sent-139, score-0.064]

66 We used a squared exponential covariance function for the prior distribution on the latent function f . [sent-140, score-0.254]

67 The inferred posterior mean function for f , together with 95% conﬁdence intervals, is shown in Figure 1(a). [sent-141, score-0.057]

68 2 Notice that the right hand tail of the inferred mean function shows an oscillatory behaviour. [sent-145, score-0.081]

69 We believe that this is an artifact caused by the squared exponential covariance; the steep rise between time zero and time two forces the length scale of the function to be small, hence giving rise to wavy functions [see page 123 in 8]. [sent-146, score-0.156]

70 To avoid this, we repeated the experiment using the “MLP” covariance function for the prior distribution over f [12]. [sent-147, score-0.099]

71 The MLP covariance is obtained as the limiting case of an inﬁnite number of sigmoidal neural networks and has the following covariance function k (t, t ) = arcsin wtt + b (wt2 + b + 1) (wt 2 + b + 1) (19) where w and b are parameters known as the weight and the bias variance. [sent-149, score-0.124]

72 The results using this covariance function are shown in Figure 1(b). [sent-150, score-0.062]

73 The resulting proﬁle does not show the unexpected oscillatory behaviour and has tighter credibility intervals. [sent-151, score-0.115]

74 The hyperparameters were assigned a vague gamma prior distribution (a = b = 0. [sent-154, score-0.069]

75 Differences in the estimates of the basal transcription rates are probably due to the different methods used for probe-level processing of the microarray data. [sent-160, score-0.52]

76 2 Non-linear response analysis We then used the non-linear response model of section 2 in order to constrain the protein concentrations inferred to be positive. [sent-162, score-0.498]

77 We achieved this by using an exponential response of the transcription rate to the logged protein concentration. [sent-163, score-0.676]

78 The inferred MAP solutions for the latent function f are plotted in Figure 3 for the squared exponential prior (a) and for the MLP prior (b). [sent-164, score-0.286]

79 4 4 3 3 2 2 1 1 0 0 −1 −1 −2 0 5 (a) −2 10 0 5 (b) 10 Figure 1: Predicted protein concentration for p53 using a linear response model: (a) squared exponential prior on f ; (b) MLP prior on f . [sent-167, score-0.595]

80 Solid line is mean prediction, dashed lines are 95% credibility intervals. [sent-168, score-0.091]

81 The bar charts show (a) Basal transcription rates from our model and that of Barenco et al. [sent-189, score-0.414]

82 Grey are estimates obtained with our model, white are the estimates obtained by Barenco et al. [sent-191, score-0.1]

83 4 Discussion In this paper we showed how Gaussian processes can be used effectively in modelling the dynamics of a very simple regulatory network motif. [sent-194, score-0.205]

84 This approach has many advantages over standard parametric approaches: ﬁrst of all, there is no need to restrict the inference to the observed time points, and the temporal continuity of the inferred functions is accounted for naturally. [sent-195, score-0.085]

85 It is well known that biological data exhibits a large variability, partly because of technical noise (due to the difﬁculty to measure mRNA abundance for low expressed genes, for example), and partly because of the difference between different cell lines. [sent-197, score-0.186]

86 Accounting for these sources of noise in a parametric model can be difﬁcult (particularly when estimates of the derivatives of the measured quantities are required), while Gaussian Processes can incorporate this information naturally. [sent-198, score-0.066]

87 Finally, MCMC parameter estimation in a discretised model can be computationally expensive due to the high correlations between variables. [sent-199, score-0.055]

88 This is a consequence of treating the protein concentrations as parameters, and results in many MCMC iterations to obtain reliable samples. [sent-200, score-0.259]

89 For example, it is well known that transcriptional delays can play a signiﬁcant role in determining the dynamics of many cellular processes [5]. [sent-203, score-0.204]

90 These effects can be introduced naturally in a Gaussian process model; however, the data must be sampled at a reasonably high frequency in order for delays to become identiﬁable in a stochastic model, which is often not the case with microarray data sets. [sent-204, score-0.104]

91 Another natural extension of our work would be to consider more biologically meaningful nonlinearities, such as the popular Michaelis-Menten model of transcription used in [9]. [sent-205, score-0.349]

92 Finally, networks consisting of a single transcription factor are very useful to study small systems of particular interest such as p53. [sent-206, score-0.399]

93 Solid line is mean prediction, dashed lines show 95% credibility intervals. [sent-208, score-0.091]

94 The results shown are for exp(f ), hence the asymmetry of the credibility intervals. [sent-209, score-0.091]

95 We gratefully acknowledge support from BBSRC Grant No BBS/B/0076X “Improved processing of microarray data with probabilistic models”. [sent-215, score-0.055]

96 Network component analysis: Reconstruction of regulatory signals in biological systems. [sent-243, score-0.124]

97 A tractable probabilistic model for affymetrix probelevel analysis across multiple chips. [sent-251, score-0.055]

98 Model based identiﬁcation of transcription factor activity from microarray data. [sent-286, score-0.454]

99 Bayesian sparse hidden components analysis for transcription regulation networks. [sent-292, score-0.389]

100 A probabilistic dynamical model for quantitative inference of the regulatory mechanism of transcription. [sent-299, score-0.122]

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Author: Joseph M. Kimmel, Richard M. Salter, Peter J. Thomas

Abstract: 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 [5]. 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 [8]. 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 [9]. 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 [3] and sensing temporal signals [4], 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 [5], 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 = 7.5µm, [12]) centered in a cubic volume (side length L = 30µm). 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 [12] 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/30µm, 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 = ±13.5µm 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 = 7.5µm; cube side L = 30µm. 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). 2.2 Analysis 2.2.1 Assumptions We make the following assumptions to simplify the analysis of the distribution of receptor activities at equilibrium, whether pre- or post-stimulus onset: 1. Independence. At equilibrium, the state of each receptor (bound vs unbound) is independent of the states of the other receptors. 2. Linear Gradient. At equilibrium under the imposed gradient condition, the concentration of ligand molecule varies linearly with position along the gradient axis. 3. Symmetry. 1 http://nwg.phy.bnl.gov/∼bviren/uno/other/ (a) Rotational equivariance of receptor activities. In the absence of an applied gradient signal, the probability distribution describing the receptor states is equivariant with respect to arbitrary rotations of the sphere. (b) Rotational invariance of gradient direction. The imposed gradient seen by a model cell is equally likely to be coming from any direction; therefore the gradient direction vector is uniformly distributed over S2 . (c) Axial equivariance about the gradient direction. Once a gradient direction is imposed, the probability distribution describing receptor states is rotationally equivariant with respect to rotations about the axis parallel with the gradient. Berg and Purcell [1] calculate the inaccuracy in concentration estimates due to nonindependence of adjacent receptors; for our parameters (effective receptor radius = 5nm, receptor spacing ∼ 1µm) the fractional error in estimating concentration differences due to receptor nonindependence is negligible ( 10−11 ) [1, 2]. Because we ﬁx receptors to be in 1:1 correspondence with surface tiles, spherical symmetry and uniform distribution of the receptors are only approximate. The gradient signal communicated via diffusion does not involve sharp spatial changes on the scale of the distance between nearby receptors, therefore spherical symmetry and uniform identical receptor distribution are good analytic approximations of the model conﬁguration. By rotational equivariance we mean that combining any rotation of the sphere with a corresponding rotation of the indices labeling the N receptors, {j = 1, · · · , N }, leads to a statistically indistinguishable distribution of receptor activities. This same spherical symmetry is reﬂected in the a priori distribution of gradient directions, which is uniform over the sphere (with density 1/4π). Spherical symmetry is broken by the gradient signal, which ﬁxes a preferred direction in space. About this axis however, we assume the system retains the rotational symmetry of the cylinder. 2.2.2 Mutual information of the receptors In order to quantify the directional information available to the cell from its surface receptors we construct an explicit model for the receptor states and the cell’s estimated direction. We model the receptor states via a collection of random variables {Bj } and develop an expression for the entropy of {Bj }. Then in section 2.2.3 we present a method for projecting a temporally ﬁltered estimated direction, g , into three (rather than N ) dimensions. ˆ N Let the random variables {Bj }j=1 represent the states of the N cAMP receptors on the cell surface; Bj = 1 if the receptor is bound to a molecule of cAMP, otherwise Bj = 0. Let xj ∈ S2 represent the direction from the center of the center of the cell to the j th receptor. Invoking assumption 2 above, we take the equilibrium concentration of cAMP at x to be c(x|g) = a + b(x · g) where g ∈ S2 is a unit vector in the direction of the gradient. The parameter a is the mean concentration over the cell surface, and b = R| c| is half the drop in concentration from one extreme on the cell surface to the other. Before the stimulus begins, the gradient direction is undeﬁned. It can be shown (see Supplemental Materials) that the entropy of receptor states given a ﬁxed gradient direction g, H[{Bj }|g], is given by an integral over the sphere: π 2π a + b cos(θ) sin(θ) dφ dθ (as N → ∞). (1) a + b cos(θ) + Keq 4π θ=0 φ=0 On the other hand, if the gradient direction remains unspeciﬁed, the entropy of receptor states is given by H[{Bj }|g] ∼ N Φ π 2π θ=0 φ=0 H[{Bj }] ∼ N Φ a + b cos(θ) a + b cos(θ) + Keq sin(θ) dφ dθ (as N → ∞), 4π − (p log2 (p) + (1 − p) log2 (1 − p)) , 0 < p < 1 0, p = 0 or 1 binary random variable with state probabilities p and (1 − p). where Φ[p] = (2) denotes the entropy for a In both equations (1) and (2), the argument of Φ is a probability taking values 0 ≤ p ≤ 1. In (1) the values of Φ are averaged over the sphere; in (2) Φ is evaluated after averaging probabilities. Because Φ[p] is convex for 0 ≤ p ≤ 1, the integral in equation 1 cannot exceed that in equation 2. Therefore the mutual information upon receiving the signal is nonnegative (as expected): ∆ M I[{Bj }; g] = H[{Bj }] − H[{Bj }|g] ≥ 0. The analytic solution for equation (1) involves the polylogarithm function. For the parameters shown in the simulation (a = 1.078 nMol, b = .512 nMol, Keq = 25 nMol), the mutual information with 980 receptors is 2.16 bits. As one would expect, the mutual information peaks when the mean concentration is close to the Keq of the receptor, exceeding 16 bits when a = 25, b = 12.5 and Keq = 25 (nMol). 2.2.3 Dimension reduction The estimate obtained above does not give tell us how quickly the directional information available to the cell evolves over time. Direct estimate of the mutual information from stochastic simulations is impractical because the aggregate random variables occupy a 980 dimensional space that a limited number of simulation runs cannot sample adequately. Instead, we construct a deterministic function from the set of 980 time courses of the receptors, {Bj (t)}, to an aggregate directional estimate in R3 . Because of the cylindrical symmetry inherent in the system, our directional estimator g is ˆ an unbiased estimator of the true gradient direction g. The estimator g (t) may be thought of as ˆ representing a downstream chemical process that accumulates directional information and decays with some time constant τ . Let {xj }N be the spatial locations of the N receptors on the cell’s j=1 surface. Each vector is associated with a weight wj . Whenever the j th receptor binds a cAMP molecule, wj is incremented by one; otherwise wj decays with time constant τ . We construct an instantaneous estimate of the gradient direction from the linear combination of receptor positions, N gτ (t) = j=1 wj (t)xj . This procedure reﬂects the accumulation and reabsorption of intracellular ˆ second messengers released from the cell membrane upon receptor binding. Before the stimulus is applied, the weighted directional estimates gτ are small in absolute magniˆ tude, with direction uniformly distributed on S2 . In order to determine the information gained as the estimate vector evolves after stimulus application, we wish to determine the change in entropy in an ensemble of such estimates. As the cell gains information about the direction of the gradient signal from its receptors, the entropy of the estimate should decrease, leading to a rise in mutual information. By repeating multiple runs (M = 600) of the simulation we obtain samples from the ensemble of direction estimates, given a particular stimulus direction, g. In the method of Kozachenko and Leonenko [13], adapted for the analysis of neural spike train data by Victor [14] (“KLV method”), the cumulative distribution function is approximated directly from the observed samples, and the entropy is estimated via a change of variables transformation (see below). This method may be formulated in vector spaces Rd for d > 1 ([13]), but it is not guaranteed to be unbiased in the multivariate case [15] and has not been extended to curved manifolds such as the sphere. In the present case, however, we may exploit the symmetries inherent in the model (Assumptions 3a-3c) to reduce the empirical entropy estimation problem to one dimension. Adapting the argument in [14] to the case of spherical data from a distribution with rotational symmetry about a given axis, we obtain an estimate of the entropy based on a series of observations of the angles {θ1 , · · · , θM } between the estimates gτ and the true gradient direction g (for details, see ˆ Supplemental Materials): 1 H∼ M M log2 (λk ) + log2 (2(M − 1)) + k=1 γ + log2 (2π) + log2 (sin(θk )) loge (2) (3) ∆ (as M → ∞) where after sorting the θk in monotonic order, λk = min(|θk − θk±1 |) is the distance between each angle and its nearest neighbor in the sample, and γ is the Euler-Mascheroni constant. As shown in Figure 2, this approximation agrees with the analytic result for the uniform distribution, Hunif = log2 (4π) ≈ 3.651. 3 Results Figure 3 shows the results of M = 600 simulation runs. Panel A shows the concentration averaged across a set of 1µm3 sample boxes, four in the x = −13.5µm plane and four in the x = +13.5µm Figure 2: Monte Carlo simulation results and information analysis. A: Average concentration proﬁles along two planes perpendicular to the gradient, at x = ±13.5µm. B: Estimated direction vector (x, y, and z components; x = dark blue trace) gτ , τ = 500 msec. C: Entropy of the ensemble of diˆ rectional vector estimates for different values of the intracellular ﬁltering time constant τ . Given the directions of the estimates θk , φk on each of M runs, we calculate the entropy of the ensemble using equation (3). All time constants yield uniformly distributed directional estimates in the pre-stimulus period, 0 ≤ t ≤ 2 (sec). After stimulus onset, directional estimates obtained with shorter time constants respond more quickly but achieve smaller gains in mutual information (smaller reductions in entropy). Filtering time constants τ range from lightest to darkest colors: 20, 50, 100, 200, 500, 1000, 2000 msec. plane. The initial bolus of cAMP released into the volume at t = 0 sec is not uniformly distributed, but spreads out evenly within 0.25 sec. At t = 2.0 sec the boundary conditions are changed, causing a gradient to emerge along a realistic time course. Consistent with the analytic solution for the mean concentration (not shown), the concentration approaches equilibrium more rapidly near the absorbing wall (descending trace) than at the imposed ﬂux wall (ascending trace). Panel B shows the evolution of a directional estimate vector gτ for a single run, with τ = 500 ˆ msec. During uniform conditions all vectors ﬂuctuate near the origin. After gradient onset the variance increases and the x component (dark trace) becomes biased towards the gradient source (g = [−1, 0, 0]) while the y and z components still have a mean of zero. Across all 600 runs the mean of the y and z components remains close to zero, while the mean of the x component systematically departs from zero shortly after stimulus onset (not shown). Hence the directional estimator is unbiased (as required by symmetry). See Supplemental Materials for the population average of g . ˆ Panel C shows the time course of the entropy of the ensemble of normalized directional estimate vectors gτ /|ˆτ | over M = 600 simulations, for intracellular ﬁltering time constants ranging from 20 ˆ g msec to 2000 msec (light to dark shading), calculated using equation (3). Following stimulus onset, entropy decreases steadily, showing an increase in information available to the amoeba about the direction of the stimulus; the mutual information at a given point in time is the difference between the entropy at that time and before stimulus onset. For a cell with roughly 1000 receptors the mutual information has increased at most by ∼ 2 bits of information by one second (for τ = 500 msec), and at most by ∼ 3 bits of information by two seconds (for τ =1000 or 2000 msec), under our stimulation protocol. A one bit reduction in uncertainty is equivalent to identifying the correct value of the x component (positive versus negative) when the stimulus direction is aligned along the x-axis. Alternatively, note that a one bit reduction results in going from the uniform distribution on the sphere to the uniform distribution on one hemisphere. For τ ≤ 100 msec, the weighted average with decay time τ never gains more than one bit of information about the stimulus direction, even at long times. This observation suggestions that signaling must involve some chemical components with lifetimes longer than 100 msec. The τ = 200 msec ﬁlter saturates after about one second, at ∼ 1 bit of information gain. Longer lived second messengers would respond more slowly to changes from the background stimulus distribution, but would provide better more informative estimates over time. The τ = 500 msec estimate gains roughly two bits of information within 1.5 seconds, but not much more over time. Heuristically, we may think of a two bit gain in information as corresponding to the change from a uniform distribution to one covering uniformly covering one quarter of S2 , i.e. all points within π/3 of the true direction. Within two seconds the τ = 1000 msec and τ = 2000 msec weighted averages have each gained approximately three bits of information, equivalent to a uniform distribution covering all points with 0.23π or 41o of the true direction. 4 Discussion & conclusions Clearly there is an opportunity for more precise control of experimental conditions to deepen our understanding of spatio-temporal information processing at the membranes of gradient-sensitive cells. Efforts in this direction are now using microﬂuidic technology to create carefully regulated spatial proﬁles for probing cellular responses [16]. Our results suggest that molecular processes relevant to these responses must have lasting effects ≥ 100 msec. We use a static, immobile cell. Could cell motion relative to the medium increase sensitivity to changes in the gradient? No: the Dictyostelium velocity required to affect concentration perception is on order 1cm sec−1 [1], whereas reported velocities are on the order µm sec−1 [5]. The chemotactic response mechanism is known to begin modifying the cell membrane on the edge facing up the gradient within two seconds after stimulus initiation [7, 6], suggesting that the cell strikes a balance between gathering data and deciding quickly. Indeed, our results show that the reported activation of the G-protein signaling system on the leading edge of a chemotactically responsive cell [7] rises at roughly the same rate as the available chemotactic information. Results such as these ([7, 6]) are obtained by introducing a pipette into the medium near the amoeba; the magnitude and time course of cAMP release are not precisely known, and when estimated the cAMP concentration at the cell surface is over 25 nMol by a full order of magnitude. Thomson and Kristan [17] show that for discrete probability distributions and for continuous distributions over linear spaces, stimulus discriminability may be better quantiﬁed using ideal observer analysis (mean squared error, for continuous variables) than information theory. The machinery of mean squared error (variance, expectation) do not carry over to the case of directional data without fundamental modiﬁcations [18]; in particular the notion of mean squared error is best represented by the mean resultant length 0 ≤ ρ ≤ 1, the expected length of the vector average of a collection of unit vectors representing samples from directional data. A resultant with length ρ ≈ 1 corresponds to a highly focused probability density function on the sphere. In addition to measuring the mutual information between the gradient direction and an intracellular estimate of direction, we also calculated the time evolution of ρ (see Supplemental Materials.) We ﬁnd that ρ rapidly approaches 1 and can exceed 0.9, depending on τ . We found that in this case at least the behavior of the mean resultant length and the mutual information are very similar; there is no evidence of discrepancies of the sort described in [17]. We have shown that the mutual information between an arbitrarily oriented stimulus and the directional signal available at the cell’s receptors evolves with a time course consistent with observed reaction times of Dictyostelium amoeba. Our results reinforce earlier claims that information theory can play a role in understanding biochemical cellular communication. Acknowledgments MCell simulations were run on the Oberlin College Beowulf Cluster, supported by NSF grant CHE0420717. References [1] Howard C. Berg and Edward M. Purcell. Physics of chemoreception. Biophysical Journal, 20:193, 1977. [2] William Bialek and Sima Setayeshgar. Physical limits to biochemical signaling. PNAS, 102(29):10040– 10045, July 19 2005. [3] S. Qazi, A. Beltukov, and B.A. Trimmer. Simulation modeling of ligand receptor interactions at nonequilibrium conditions: processing of noisy inputs by ionotropic receptors. Math Biosci., 187(1):93–110, Jan 2004. [4] D. J. Spencer, S. K. Hampton, P. Park, J. P. Zurkus, and P. J. Thomas. The diffusion-limited biochemical signal-relay channel. In S. Thrun, L. Saul, and B. Sch¨ lkopf, editors, Advances in Neural Information o Processing Systems 16. MIT Press, Cambridge, MA, 2004. [5] P.R. Fisher, R. Merkl, and G. Gerisch. Quantitative analysis of cell motility and chemotaxis in Dictyostelium discoideum by using an image processing system and a novel chemotaxis chamber providing stationary chemical gradients. J. Cell Biology, 108:973–984, March 1989. [6] Carole A. Parent, Brenda J. Blacklock, Wendy M. Froehlich, Douglas B. Murphy, and Peter N. Devreotes. G protein signaling events are activated at the leading edge of chemotactic cells. Cell, 95:81–91, 2 October 1998. [7] Xuehua Xu, Martin Meier-Schellersheim, Xuanmao Jiao, Lauren E. Nelson, and Tian Jin. Quantitative imaging of single live cells reveals spatiotemporal dynamics of multistep signaling events of chemoattractant gradient sensing in dictyostelium. Molecular Biology of the Cell, 16:676–688, February 2005. [8] Jan Wouter-Rappel, Peter. J Thomas, Herbert Levine, and William F. Loomis. Establishing direction during chemotaxis in eukaryotic cells. Biophys. J., 83:1361–1367, 2002. [9] T.M. Cover and J.A. Thomas. Elements of Information Theory. John Wiley, New York, 1990. [10] J. R. Stiles, D. Van Helden, T. M. Bartol, E.E. Salpeter, and M. M. Salpeter. Miniature endplate current rise times less than 100 microseconds from improved dual recordings can be modeled with passive acetylcholine diffusion from a synaptic vesicle. Proc. Natl. Acad. Sci. U.S.A., 93(12):5747–52, Jun 11 1996. [11] J. R. Stiles and T. M. Bartol. Computational Neuroscience: Realistic Modeling for Experimentalists, chapter Monte Carlo methods for realistic simulation of synaptic microphysiology using MCell, pages 87–127. CRC Press, Boca Raton, FL, 2001. [12] M. Ueda, Y. Sako, T. Tanaka, P. Devreotes, and T. Yanagida. Single-molecule analysis of chemotactic signaling in Dictyostelium cells. Science, 294:864–867, October 2001. [13] L.F. Kozachenko and N.N. Leonenko. Probl. Peredachi Inf. [Probl. Inf. Transm.], 23(9):95, 1987. [14] Jonathan D. Victor. Binless strategies for estimation of information from neural data. Physical Review E, 66:051903, Nov 11 2002. [15] Marc M. Van Hulle. Edgeworth approximation of multivariate differential entropy. Neural Computation, 17:1903–1910, 2005. [16] Loling Song, Sharvari M. Nadkarnia, Hendrik U. B¨ dekera, Carsten Beta, Albert Bae, Carl Franck, o Wouter-Jan Rappel, William F. Loomis, and Eberhard Bodenschatz. Dictyostelium discoideum chemotaxis: Threshold for directed motion. Euro. J. Cell Bio, 85(9-10):981–9, 2006. [17] Eric E. Thomson and William B. Kristan. Quantifying stimulus discriminability: A comparison of information theory and ideal observer analysis. Neural Computation, 17:741–778, 2005. [18] Kanti V. Mardia and Peter E. Jupp. Directional Statistics. 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