PARAMETER UNCERTAINTY QUANTIFICATION USING SURROGATE MODELS APPLIED TO A SPATIAL MODEL OF YEAST MATING POLARIZATION

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1 Ching-Shan Chou Department of Mathematics Ohio State University PARAMETER UNCERTAINTY QUANTIFICATION USING SURROGATE MODELS APPLIED TO A SPATIAL MODEL OF YEAST MATING POLARIZATION

2 In systems biology, we would like to Develop a comprehensive model that captures the key features in the modeled system Identify the key processes in the system Estimate the parameters in the model so that the model has a predictive power

3 The challenges are With a (relatively) comprehensive model, there are a lot of unknown parameters Experimental data are generated in order to estimate the parameters, but Is the system identifiable? Do we have enough data? Are the data useful? How to deal with spatial data?

Budding yeast (Saccharomyces cerevisiae) as a model system Genome fully sequenced (1996) Simple geometry Pathway understood extensively Cdc42 GTPase, the central

4 Budding yeast (Saccharomyces cerevisiae) as a model system Genome fully sequenced (1996) Simple geometry Pathway understood extensively Cdc42 GTPase, the central polarization protein, is highly conserved (human Cdc42 (Cdc42hs) is 8% identical to yeast Cdc42 ) Image credit: Masur - Own work. Licensed under Public domain via Wikimedia Commons

5 Cell cycles of budding yeast

6 Yeast cell polarization Yeast reproduces by budding or mating. During mating, cell forms a projection in response to a pheromone signal. Polarization occurs via a signaling pathway with positive and negative feedback. Mogilner et al, Science 212

7 Cell polarization is everywhere Cell polarization is the asymmetric organization of cellular components. Anda et al, Journal of Cell Science 28 Fundamental to cell differentiation, division, migration, specialized functions, Etienne-Manneville, Journal of Cell Science 24

8 Yeast mating projection Shmoo

9 Accurate mating

10 α-factor induced yeast cell projection α-factor receptor (Ste2) a α α-factor

11 α-factor induced yeast cell projection α-factor GTP α Cell Cycle Arrest Cell Polarization P Transcription GTP GDP β α γ β γ

12 All-or-none polarization of proteins α-factor gradient Travis Moore, Ching-Shan Chou, Qing Nie, Noo Li Jeon and Tau-Mu Yi, Robust Spatial Sensing of Mating Pheromone Gradients by Yeast Cells, PLoS ONE, 28.

13 Signal transduction pathway

14 Signaling transduction pathway L L L Cdc24m Cdc42 Cdc42a Bem1m Cla4 L Gα Gβγ Bem1 G L L Cdc24 membrane cytoplasm

15 Signaling pathway Cdc42 Cdc24m Cdc42 is activated by Cdc24m Cdc42a can deactivate to reform Cdc42 Proteins diffuse along the membrane Cdc42a [C42a] t = D./1 [C42a] + k /1 C24m C42 k /7 C42a diffusion activation by Cdc24m deactivation

16 Computational domain z α z

17 Full polarization model Large nonlinear PDE expensive to solve! 35 unknown parameters

18 Full polarization model Quantity of Interest: Extent of polarization of Cdc42a at steady state (how concentrated is active Cdc42?) How much Cdc42 is activated? PF =.745 PF =.5

19 Introduction How do parameters affect model output? Model f(x, θ) + Parameters θ How do we minimize computational cost? Data Can data inform choice of parameters?

20 Parameter sensitivity Question: Can we quantify the effects of parameters on the behavior of the system? Let z be a scalar quantity of interest, Local sensitivity: S j = z θ j at some point z = f(θ 1, θ 2, θ N ) Can be computed analytically or approximated by finite difference Only gives information at one point what if we don t know the parameter values? θ

Global sensitivity Parameter sensitivity Correlations Pearson correlation coefficient Rank/Spearman correlation coefficient PRCC (accounts for effects of other parameters) Variance-based sensitivity

21 Global sensitivity Parameter sensitivity Correlations Pearson correlation coefficient Rank/Spearman correlation coefficient PRCC (accounts for effects of other parameters) Variance-based sensitivity measures Variance decomposition N Var(z) = V i + V ij V 1,2,...,N i=1 i< j Main effect index V j Var(z), Total-effect index Derivative-based sensitivity measures E z θ j, E z θ j, E ( z θ j ) 2

22 Methods: Parameter estimation Question : Given experimental data, can we estimate the parameters? What are the most likely parameter values? How much uncertainty is there in our parameter estimates? 4 Bayesian inference allows us to recover a probability distribution of the parameters. Probability density High uncertainty Low uncertainty Parameter value

Methods: Parameter estimation Markov chain Monte Carlo (Metropolis-Hastings): 1. Choose initial parameter set θ G. 2. For n =, 1, 2, i. Sample a new parameter set θ close to θ K ii.

23 Methods: Parameter estimation Markov chain Monte Carlo (Metropolis-Hastings): 1. Choose initial parameter set θ G. 2. For n =, 1, 2, i. Sample a new parameter set θ close to θ K ii. Calculate the acceptance ratio α = L(M ) L(M N ). iii. Accept or reject Generate a uniform random number u on [,1] If u α, set θ KQR = θ (accept). If u > α, set θ KQR = θ K (reject). Florian Hartig, theoreticalecology.wordpress.com/

24 How do we sample a high-dimensional parameter space when each sample requires an expensive model evaluation (e.g. a large PDE)? Short answer: We don t. Methods: Polynomial surrogate Solution We can fit a polynomial to z in terms of the parameters θ R, θ,, θ T. z P(θ R, θ,, θ T ) Polynomial basis depends on probability distribution of the parameters. The polynomial serves as a surrogate for the full model.

25 Generalized Polynomial Chaos (gpc) Suppose θ X ~ U[ 1,1] for all i = 1,, N. For i = 1,, N, let φ ] θ X R 7 ]^G be a set of orthogonal polynomials, i.e. 1 2 _ φ ] θ X φ K θ X dy X = h ] δ ]K dr For the uniform distribution, this gives the Legendre polynomials. These form a basis for univariate polynomials up to degree d.

26 Generalized Polynomial Chaos (gpc) The corresponding N-variate orthogonal polynomial space is defined to be W T f = d if (span φ ] θ X 7 n ]^G ) where d = (d R, d,, d T ) is a multi-index with d = T X^R d X. The Pth order gpc approximation of z can be obtained by projecting z onto the space W f T : s z T f θ = p z ]Φ ] (θ) ]^R Options: Orthogonal projection (Galerkin), polynomial fitting

27 Polynomial fitting s z T f θ = p z ]Φ ] (θ) ]^R Want to solve for polynomial coefficients, given samples θ (R),, θ (t). Let x = (z R,, z ŝ ) be the vector of coefficients. Ax = b A is a matrix where each row corresponds to one sample and each column corresponds to one basis polynomial. b is a vector of model output at the sample points.

28 Polynomial fitting Dimension of polynomial space is TQf T, where P is the degree. # of parameters Polynomial degree # of coefficients Polynomial fitting: Undersampling Exact fitting Oversampling Compressed sensing Interpolation Least squares approximation

29 Least squares approximation # of samples > # of coefficients A x = b Generally no exact solution Find polynomial that minimizes the squared error, K p P θ X b X X^R Equivalent to solving A x Ax = A x b

30 Compressed sensing (in a nutshell) # of samples < # of coefficients A x = b Infinitely many solutions Find solution with minimal l1-norm (the sparsest polynomial)

31 Methods: polynomial surrogate Now that we have a polynomial surrogate, S z = E } M ~ can be analytically computed no sampling required. Markov chain Monte Carlo (MCMC) samples require only a polynomial evaluation. The only model evaluations required are to fit the polynomial.

32 Methods: summary Determine quantity of interest Fit polynomial* Evaluate polynomial error Additional samples Satisfactory? No Yes Perform sensitivity analysis Can parameter count be reduced? No Yes Fix non-sensitive parameters Parameter estimation

33 Results: Toy model Toy model: ODE model of heterotrimeric G-protein cycle R + L ODE easy to solve. 8 parameters: 6 have been measured in experiments, 2 have been estimated via optimization *. * Yi TM, Kitano H, Simon MI. A quantitative characterization of the yeast heterotrimeric G protein cycle. Proc Natl Acad Sci USA. 23;1(19):

34 Toy model Data Data output: Fraction of free Gβγ at various time and α-factor

35 Toy model Fitting with 2 parameters

36 Toy model Results Mean sensitivity k Ga k Gd Estimated MSE = 4.1 x 1-4 Free Gbg Time-course ODE solution data polynomial Free Gbg Dose-response Time (s) Alpha factor (nm) Optimal 1 MSE = 1.3 x 1-4 Free Gbg Time (s) Free Gbg Alpha factor (nm) * Yi TM, Kitano H, Simon MI. A quantitative characterization of the yeast heterotrimeric G protein cycle. Proc Natl Acad Sci USA. 23;1(19):

37 Toy model Approximating all 8 parameters Approximating all 8 parameters: 5 th degree polynomial with 15 samples Error: Mean.221, Std Data: Fraction of free Gβγ at various time points, α-factor levels

38 Toy model 8 parameter MCMC results k RL k Rs k RLm k Rd Estimated MSE = 6.4 x 1-4 Free Gbg Time-course ODE solution data polynomial Time (s) Free Gbg Dose-response Alpha factor (nm) k Ga k Rd k G k Gd Optimal MSE = 1.3 x 1-4 Free Gbg Time (s) Free Gbg Alpha factor (nm)

39 Findings System is most sensitive to activation and deactivation of Gα Polynomial surrogates produce similar results to optimization method Sensitive parameters have lower uncertainty in estimation High uncertainty occurs in parameters that have been experimentally measured

40 Full polarization model Large nonlinear PDE expensive to solve! 35 unknown parameters

41 Full polarization model Polynomial fitting 5 th order polynomial 658,8 coefficients Undersample and use l1-minimization With 5 samples, mean absolute error was Frequency Error.2.4

42 Full polarization model Sensitivity Most parameters have small sensitivity. Reduce parameter count!

43 Full polarization model 15 parameters k 24cm 1.5 Fit new polynomial in reduced 1.5 space with 6 samples. 4x1-3 4x1-1 k 42a k B1mc C24 t x 1 3 h k 24cm x x1-2 k 42d x1-2 2 k Cla4a x1-4 6x1-2 B1 t x1 3 5x1 3 D c x1-3 2x1-2 k 24d k B1cm k Cla4d q D c42a x1-3 2x1-2

44 Full polarization model Estimated parameters Polynomial surrogates can also be applied to optimization methods, e.g. simulated annealing (SA). 5 4 data MCMC SA SA produces better fit. MCMC shows that we must be careful with interpretation. optimal parameter sets are not unique. C42a π -π/2 π/2 π Angle from front of cell

45 What we ve done: Summary 1. Fit polynomial surrogate using 5 samples 2. Parameter sensitivity analysis to reduce parameter count 3. Fit new polynomial in reduced space with 6 samples 4. Parameter estimation via MCMC (chain length 2,,) Cost of estimation: 11, model evaluations vs. 2,, 18-fold reduction in cost Parallelizable

46 Conclusions System is most sensitive to parameters associated with Cdc42 cycle and feedback loops. Current data is not enough to estimate parameters, but indicates difference between active and inactive Cdc42. Can use time series data, other protein species. Problems: time delay, correlation between species. Addition of other protein species Model selection.

47 Acknowledgement Marissa Renardy (OSU) Tau-Mu Yi (UCSB) Dongbin Xiu (OSU) CAREER DMS

Modeling Yeast Cell Polarization Induced by Pheromone Gradients

Journal of Statistical Physics ( C 27 ) DOI: 1.17/s1955-7-9285-1 Modeling Yeast Cell Polarization Induced by Pheromone Gradients Tau-Mu Yi, 1 Shanqin Chen, 2 Ching-Shan Chou 2 and Qing Nie 2 Received March