Intrinsic Noise in Nonlinear Gene Regulation Inference Chao Du Department of Statistics, University of Virginia Joint Work with Wing H. Wong, Department of Statistics, Stanford University
Transcription Regulation Regulation Gene 1 Transcription Translation RNA 1 Protein 1 Gene 2 Transcription Translation RNA 2 Protein 2 Gene n Transcription RNA n Translation Protein n
Transcription Regulation Regulation Gene 1 Gene 2 Transcription Transcription Translation RNA 1 Protein 1 Translation RNA 2 Protein 2 Logical Models Boolean network Probabilistic Boolean networks Gene n Transcription RNA n Translation Protein n Continuous Models Dynamic Bayesian network Linear differential equations
Physical-Oriented Gene Regulation Models Nonlinear: The regulation system is nonlinear, as dictated by the laws of physics Intrinsic Noise: Unlike the traditional ensemble experiments, modern single cell measurement technology allows us to observe the cell to cell variation, known as intrinsic noise. Thus, regulation system is intrinsically stochastic and discrete.
Physical-Oriented Gene Regulation Models Nonlinear: The regulation system is nonlinear, as dictated by the laws of physics Intrinsic Noise: Unlike the traditional ensemble experiments, modern single cell measurement technology allows us to observe the cell to cell variation, known as intrinsic noise. Thus, regulation system is intrinsically stochastic and discrete. Ensemble Average Measurements Single-Cell Level Measurements X 1 X 1 X 2 X 3 X 4 X X 2 X 3 X 4. X 1 X 2 X 3 X 4
Gene Regulation Model and Assumption Transcription if i (y) x i RNA i RNA i x i Translation r i x i y i Protein i P rotein i y i
Gene Regulation Model and Assumption Transcription if i (y) x i RNA i RNA i x i Translation r i x i y i Protein i P rotein i y i Quasi-Steady-State Assumption As RNA metabolism is often much slower (or faster) than protein metabolism, we may assume that the concentrations of protein (or RNA) species always reach quasi-steady state instantly. Production i f i (x) x i Gene i ix i
Gene Regulation Model and Assumption Production Rate as Nonlinear Rational Function Production i f i (x) x i Gene i i f i (x) = i b i0 + P m i b ijs ij 1+ P m i c ijs ij ix i S ij = Y k x n kij k 0 apple b i0 apple 1 0 apple b ij apple c ij
Gene Regulation Model and Assumption Production Rate as Nonlinear Rational Function Production i f i (x) x i Gene i i f i (x) = i b i0 + P m i b ijs ij 1+ P m i c ijs ij ix i S ij = Y k x n kij k 0 apple b i0 apple 1 0 apple b ij apple c ij f 1 (x) = 0.5+x 1 +0.01x 2 x 3 1+x 1 +0.9x 2 2 +0.2x 2x 3 0.5 1 x 1 x 1 0 0.9x 2 2 0.01x 2 x 3 0.2x 2 x 3 Gene 3 NA P1 P2 P2 P2 P3 Gene 1 Gene 2 NA Activator Repressor Repressor
Modeling the Intrinsic Noise Model the system with a multivariate birth-death process X = {X 1,X 2,,X M } Production i F i (X) X i Gene i ix i
Modeling the Intrinsic Noise Model the system with a multivariate birth-death process X = {X 1,X 2,,X M } The evolution of the system can then be described with Kolmogorov forward equation (master equation) Production i F i (X) X i Gene i ix i dp (X) dt = X X 0 W (X! X 0 )P (X)+ X X 0 W (X 0! X)P (X 0 ) X X 0 W (X! X 0 ) {X 1,,X i +1,,X M } i F (X) {X 1,,X i,,x M } {X 1,,X i 1,,X M } ix i
How to Make Inference? Single-cell Measurements Nonlinear Stochastic System X 1 X 2 X 3 X 4 X 1 X 2 X 3 X 4. Production i F i (X) X i Gene i ix i? Inference of unknown parameters b ij =? c ij =? F i (X) = b i0 + P m i b ijs ij 1+ P m i c ijs ij S ij = Y k X n kij k
How to Make Inference? Single-cell Measurements Nonlinear Stochastic System X 1 X 2 X 3 X 4 X 1 X 2 X 3 X 4. Production i F i (X) X i Gene i ix i? Inference of unknown parameters b ij =? c ij =? F i (X) = b i0 + P m i b ijs ij 1+ P m i c ijs ij S ij = Y k X n kij k Construct linear functions of the unknown parameters!
System with One Gene Consider a system with only one gene, denoted as X. W (X! X + 1) = F (X) W (X! X 1) = X F (X) = b 0 + P N b jx j 1+ P N c jx j
System with One Gene Consider a system with only one gene, denoted as X. W (X! X + 1) = F (X) W (X! X 1) = X F (X) = b 0 + P N b jx j 1+ P N c jx j We have the identity: F (X)(1 + NX c j X j )=b 0 + NX b j X j
System with One Gene Consider a system with only one gene, denoted as X. W (X! X + 1) = F (X) W (X! X 1) = X F (X) = b 0 + P N b jx j 1+ P N c jx j We have the identity: F (X)(1 + NX c j X j )=b 0 + NX b j X j E F (X) + NX c j E F (X)X j = b 0 + NX b j E X j
System with One Gene Consider a system with only one gene, denoted as X. W (X! X + 1) = F (X) W (X! X 1) = X F (X) = b 0 + P N b jx j 1+ P N c jx j We have the identity: F (X)(1 + NX c j X j )=b 0 + NX b j X j E F (X) + NX c j E F (X)X j = b 0 + NX b j E X j
Kolmogorov Forward Equation at Steady State At the steady state, the Kolmogorov forward equation dp (X) dt =0 E (H(X 0 ) H(X))W (X! X 0 ) =0
Kolmogorov Forward Equation at Steady State At the steady state, the Kolmogorov forward equation dp (X) dt =0 E (H(X 0 ) H(X))W (X! X 0 ) =0 For different H(X), we have: H(X) =X E F (X) = E X
Kolmogorov Forward Equation at Steady State At the steady state, the Kolmogorov forward equation dp (X) dt =0 E (H(X 0 ) H(X))W (X! X 0 ) =0 For different H(X), we have: H(X) =X E F (X) = E X H(X) =X 2 E F (X)X] = E X 2 X H(X) =X N+1 E F (X)X N = E N X C j N ( 1)j X N+1 j = E g N (X)
Kolmogorov Forward Equation at Steady State At the steady state, the Kolmogorov forward equation dp (X) dt =0 E (H(X 0 ) H(X))W (X! X 0 ) =0 For different H(X), we have: H(X) =X E F (X) = E X H(X) =X 2 E F (X)X] = E X 2 X H(X) =X N+1 E F (X)X N = E N X C j N ( 1)j X N+1 j = E g N (X) H(X) =X j e X e E F (X)X j e X = E g j (X)e X
System with Multiple Genes Extend the previous method to multiple gene system:
System with Multiple Genes Extend the previous method to multiple gene system: i e i E F i (X)X N i i ( Y X N k k )e P k kx k i E g Ni (X i )( Y X N k k )e P k kx k k6=i k6=i
System with Multiple Genes Extend the previous method to multiple gene system: i e i E F i (X)X N i i ( Y X N k k )e P k kx k i E g Ni (X i )( Y X N k k )e P k kx k k6=i k6=i Which allows us to construct linear equations of the unknown parameters: F i (X) = b i0 + P m i b ijs ij 1+ P m i c ijs ij S ij = Y k X n kij k i he X i e P Xm i k kx k + i = i e hb i0 E e P Xm i k kx k + c ij E g nij (X i )S ij X n ij i e P k kx k i b ij E S ij e P k kx k i By using different sets of γi, we can have as many linear equations as we need.
Summary of Inference Procedure 1. Propose candidate model (rational function) F i (X) = b i0 + P m i b ijs ij 1+ P m i c ijs ij S ij = Y k X n kij k 2. Collect single-cell expression data from different steady states (via perturbation experiments) Gene 1 Gene 2 Cell 1 X11 (1) X21 (1) Cell 2 X12 (1) X22 (1) Cell 3 X13 (1) X23 (1)
Summary of Inference Procedure 1. Propose candidate model (rational function) F i (X) = b i0 + P m i b ijs ij 1+ P m i c ijs ij S ij = Y k X n kij k + 2. Collect single-cell expression data from different steady states (via perturbation experiments) Gene 1 Gene 2 Cell 1 X11 (1) X21 (1) Cell 2 X12 (1) X22 (1) Cell 3 X13 (1) X23 (1) i i e hê Xi e P k Xm i kx k + i hb i0 Ê e P k kx k c ij Ê g nij (X i )S ij X n ij i e P k kx k i Xm i b ij Ê S ij e P k kx k i 3. Estimate the moments of gene expression at steady states, construct linear functions of unknown parameters.
Summary of Inference Procedure 1. Propose candidate model (rational function) F i (X) = b i0 + P m i b ijs ij 1+ P m i c ijs ij S ij = Y k X n kij k + 2. Collect single-cell expression data from different steady states (via perturbation experiments) Gene 1 Gene 2 Cell 1 X11 (1) X21 (1) Cell 2 X12 (1) X22 (1) Cell 3 X13 (1) X23 (1) i i e hê Xi e P k Xm i kx k + i hb i0 Ê e P k kx k c ij Ê g nij (X i )S ij X n ij i e P k kx k i Xm i b ij Ê S ij e P k kx k i ˆbi0,, ˆb ij,, ĉ ij 3. Estimate the moments of gene expression at steady states, construct linear functions of unknown parameters. 4. Estimate unknown parameters using convex optimization.
Example: Genetic Toggle Switch Genetic toggle switch: a bistable gene-regulation network Can be formed by two repressor genes X1 X2
Example: Genetic Toggle Switch Genetic toggle switch: a bistable gene-regulation network Can be formed by two repressor genes X1 X2 Production F 1 (X 1,X 2 ) = 0.5 1+X 2 2 Production F 2 (X 1,X 2 ) = 0.5 1+X 2 1 Gene 1 X 1 Gene 2 X 2 0.01X 1 0.01X 2
Example: Genetic Toggle Switch Genetic toggle switch: a bistable gene-regulation network Can be formed by two repressor genes X1 X2 Production F 1 (X 1,X 2 ) = 0.5 1+X 2 2 Production F 2 (X 1,X 2 ) = 0.5 1+X 2 1 Gene 1 X 1 Gene 2 X 2 0.01X 1 0.01X 2
Example: Genetic Toggle Switch True Model F 1 (X 1,X 2 )= 0.5 1+X 2 2 Candidate F 1 (X 1,X 2 )= b 10 + b 11 X 1 + b 12 X 2 + b 13 X 2 1 + b 14 X 1 X 2 + b 15 X 2 2 1+c 11 X 1 + c 12 X 2 + c 13 X 2 1 + c 14X 1 X 2 + c 15 X 2 2
Example: Genetic Toggle Switch True Model F 1 (X 1,X 2 )= 0.5 1+X 2 2 Candidate F 1 (X 1,X 2 )= b 10 + b 11 X 1 + b 12 X 2 + b 13 X 2 1 + b 14 X 1 X 2 + b 15 X 2 2 1+c 11 X 1 + c 12 X 2 + c 13 X 2 1 + c 14X 1 X 2 + c 15 X 2 2 Sample from two steady-states: Unperturbed Steady-State Perturbed Steady-State (X 1,X 2 ) (X 1,X 2 = C)
b 10 b 11 b 12 b 13 b 14 b 15 c 11 c 12 c 13 c 14 c 15 True Model F 1 (X 1,X 2 )= 0.5 1+X 2 2 Candidate F 1 (X 1,X 2 )= b 10 + b 11 X 1 + b 12 X 2 + b 13 X 2 1 + b 14 X 1 X 2 + b 15 X 2 2 1+c 11 X 1 + c 12 X 2 + c 13 X 2 1 + c 14X 1 X 2 + c 15 X 2 2
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