Statistical Physics approach to Gene Regulatory Networks

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1 Statistical Physics approach to Gene Regulatory Networks Jonathan Fiorentino Supervisors: Andrea Crisanti, Andrea De Martino February 16, 2017 Scuola di dottorato Vito Volterra, XXXI ciclo

2 1 Introduction 2 Information flow in GRNs with extrinsic noise 3 Inference of critical variables from transcriptome data Jonathan Fiorentino Statistical Physics approach to GRNs February 16, / 13

3 Introduction The regulation of gene expression Gene expression = Transcription factors (TF): Proteins that bind to the promoter of a gene, activating or repressing its expression. Each TF can regulate multiple genes, each gene can be regulated by multiple TFs Gene Regulatory Network (GRN). TF 1 TF 2 TF 3... TF N ACTIVATION REPRESSION GENE 1 GENE 2 GENE 3 GENE 4... GENE M Jonathan Fiorentino Statistical Physics approach to GRNs February 16, / 13

4 Introduction Gene expression is noisy Stochasticity in gene expression: Intrinsic noise: Low copy numbers effects and diffusive motion for the recognition of the target by the regulator. Extrinsic noise: External factors influencing the parameters of Gene Regulatory Networks. Consequences Physical limits to the reliability of the regulation of gene expression. Cell-to-cell variability of gene expression. Jonathan Fiorentino Statistical Physics approach to GRNs February 16, / 13

5 Introduction Extrinsic noise is not negligible Experimental measurements in single cells The influence of intrinsic and extrinsic noise sources on the expression of a given gene is of the same order of magnitude. Elowitz M. B., Levine A. J., Siggia E. D. and Swain P. S., Stochastic gene expression in a single cell, Science, 297(5584), (2002) Raser J. M. and O shea E. K., Control of stochasticity in eukaryotic gene expression Science, 304(5678), (2004) Pedraza J. M. and van Oudenaarden A., Noise propagation in gene networks, Science, 307(5717), (2005) Hill function: Simplest model of gene regulation. ḡ i (c) = g max c n i c n i + K n i i Concentration of the TF: c [0, c max]. Mean expression level of gene i: ḡ i [0, g max]; Threshold K i : ḡ i (K i ) = g max/2; Cooperativity n i. Extrinsic noise: Parameters c max, g max and {K i } vary from cell to cell. Jonathan Fiorentino Statistical Physics approach to GRNs February 16, / 13

6 Information flow in GRNs with extrinsic noise Optimization of information flow - State of the art INPUT c :TF concentration c [0,c max ] P TF (c):probability distribution of c ACTIVATION c, P TF (c) TF (INPUT) OUTPUT g i :Expression level of gene i g i [0,1] P(g i ):Probability distribution of g i σ 2 i (c):intrinsic noise on REPRESSION g i P(g 1 c) P(g 2 c) P(g 3 c) P(g 4 c) P(g M c)... g 1, P(g 1 ), σ 2 1 (c) g 2, P(g 2 ), σ 2 2 (c) g 3, P(g 3 ), σ 2 3 (c) g 4, P( g 4 ), σ 2 4 (c)... g M, P(g M ), σ 2 M (c) GENES (OUTPUT) Tkačik G., Walczak A. M. and Bialek W., Optimizing information flow in small genetic networks,physical Review E 80.3 (2009), p Jonathan Fiorentino Statistical Physics approach to GRNs February 16, / 13

7 Information flow in GRNs with extrinsic noise Optimization of information flow - State of the art TF (INPUT) ACTIVATION REPRESSION ḡ i (c)= cn i c, P c n i +K n i ḡ i (c)= K i TF (c) i c n n i +K i i ḡ i ḡ i 1/2 1/2 n i P(g 1 c) K i c Hill functions g i : Mean expression level of gene i K i : Threshold g i (K i )=1/2 n i : Cooperativity K i P(g M c) c... g 1, P(g 1 ), σ 2 1 (c) g 2, P(g 2 ), σ 2 2 (c) g 3, P(g 3 ), σ 2 3 (c) g 4, P( g 4 ), σ 2 4 (c)... g M, P(g M ), σ 2 M (c) GENES (OUTPUT) Tkačik G., Walczak A. M. and Bialek W., Optimizing information flow in small genetic networks,physical Review E 80.3 (2009), p Jonathan Fiorentino Statistical Physics approach to GRNs February 16, / 13

8 Information flow in GRNs with extrinsic noise Measuring the efficiency of GRNs INPUT CHANNEL OUTPUT I, P(I) P(O I) O, P(O) Quantitative measure of the control power of the input on the output: Mutual information I (I, O) = di do P(I, O) log 2 P(I, O) P(I)P(O) bits (1) Maximization of I (I, O) on the input distribution gives the optimal mutual information for a fixed setting of the kinetic parameters. Jonathan Fiorentino Statistical Physics approach to GRNs February 16, / 13

9 Information flow in GRNs with extrinsic noise Optimization of information flow - State of the art Maximization of information transmission: ( δ I ({g i }; c) λ δp TF (c) ) dc P TF (c) = 0. (2) I = log 2 Z; Z: Partition function. (3) Optimization of I on {K i, n i } provides optimal network architectures. Fundamental question: Optimization of information flow can be assumed as a first principle for a theory of biological networks? Only intrisic noise sources have been considered. Tkačik G., Walczak A. M. and Bialek W., Optimizing information flow in small genetic networks, Physical Review E 80.3 (2009), p Jonathan Fiorentino Statistical Physics approach to GRNs February 16, / 13

10 Information flow in GRNs with extrinsic noise Modeling the role of extrinsic noise Thresholds {K i } M i=1 extracted from probability distributions {P(K i)} M i=1 Disordered system. Analytical approach: Computation of the annealed average I ann = log 2 Z = log 2 M i=1 dk i P(K i )Z (4) using field theory techniques. Numerical approach: Computation of the quenched average I = log 2 Z (5) Jonathan Fiorentino Statistical Physics approach to GRNs February 16, / 13

11 Inference of critical variables from transcriptome data Single-cell transcriptome data Jonathan Fiorentino Statistical Physics approach to GRNs February 16, / 13

12 Inference of critical variables from transcriptome data Noise in Multiple Sclerosis Multiple Sclerosis: Multifactorial autoimmune disease of the Central Nervous System. Etiology still largely unknown. Hypothesis Cell-to-cell variability of gene expression may contribute to the development of Multiple Sclerosis. Dataset Single-cell transcriptome data of 8 patients; 3 healthy-ill twin pairs; 1 healthy-ill control pair. Problem: Extraction of the set of critical transcripts from the dataset. Jonathan Fiorentino Statistical Physics approach to GRNs February 16, / 13

13 Inference of critical variables from transcriptome data Definition of a measure of variability Clustering of a dataset of M variables Labels s = (s (1),..., s (M) ). M k s = δ s (i),s Size of cluster s (6) i=1 m k = δ k,ks Number of clusters of size k (7) s Definition H[S] = s k s M log k s 2 M Resolution (8) H[K] = k km k M log km k 2 M Relevance (9) Haimovici A. and Marsili M., Criticality of mostly informative samples: a Bayesian model selection approach, Journal of Statistical Mechanics: Theory and Experiment (2015), P Jonathan Fiorentino Statistical Physics approach to GRNs February 16, / 13

14 Inference of critical variables from transcriptome data Definition of a measure of variability Extreme cases: H[S] = s k s M log k s 2 M ; H[K] = k 1 cluster of size M: H[S] = 0, H[K] = 0. M clusters of size 1: H[S] = log 2 M, H[K] = 0. km k M log km k 2 M H[K] theoretical upper bound Fixed M and H[S]: finding the optimal partition of data Maximization of H[K]. H[K] (bits) H[K]=H[S] H[K]<H[S] 0 H[S] (bits) log 2 M Jonathan Fiorentino Statistical Physics approach to GRNs February 16, / 13

15 Inference of critical variables from transcriptome data Results Development of a stochastic optimization algorithm based on the maximization of H[K]; Result: Extraction of a list of 40 transcripts critical for Multiple Sclerosis, currently under experimental validation. We are collaborating with: Dr. Quin Wills, Wellcome Trust Centre for Human Genetics (WTCHG), University of Oxford; Prof. Marco Salvetti and his research team, Department of Neurology and Centro Neurologico Terapia Sperimentale (CENTERS), Ospedale S. Andrea, University of Rome La Sapienza; Prof. Francesca Grassi, Department of Physiology and Pharmacology, University of Rome La Sapienza; Jonathan Fiorentino Statistical Physics approach to GRNs February 16, / 13