Presentation ECMTB14
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The document discusses parameter estimation for stochastic biological systems, particularly focusing on the inverse omega square method and its advantages in estimating parameters more accurately than traditional methods. It emphasizes the influence of molecule number fluctuations, computational complexities, and volume on estimation errors. The research provides insights into the efficacy of higher-order system size expansions for improving model accuracy in biological simulations.
Presentation ECMTB14
- 1. ECTMB Gothenburg, June 19th 2014 Parameter Estimation for the Inverse Omega Square Method Fabian Froehlich and Jan Hasenauer Parameter Estimation for the System Size Expansion Fabian Fröhlich1,2, Fabian Theis1,2, Jan Hasenauer1,2 1 Helmholtz Centre Munich, Institute for Computational Biology 2Technical University Munich, Department for Mathematics Parameter Estimation for the Inverse Omega Square Method Fabian Froehlich and Jan Hasenauer
- 2. Elowitz et. al, 20XX Taniguchi et. al, 2010 Noise in Biological Systems
- 3. Elowitz et. al, 20XX Taniguchi et. al, 2010 Noise in Biological Systems Extrinsic noise varation in external factors induces fluctuations
- 4. Elowitz et. al, 20XX Taniguchi et. al, 2010 Noise in Biological Systems Intrinsic noise small-number of molecules induces fluctuations Extrinsic noise varation in external factors induces fluctuations
- 5. Elowitz et. al, 20XX Taniguchi et. al, 2010 Noise in Biological Systems Intrinsic noise small-number of molecules induces fluctuations Extrinsic noise varation in external factors induces fluctuations
- 6. Elowitz et. al, 20XX Taniguchi et. al, 2010 Noise in Biological Systems Intrinsic noise small-number of molecules induces fluctuations Extrinsic noise varation in external factors induces fluctuations How can small molecule numbers induce stochasticity ?
- 7. Fluctuations in Molecule Numbers time large volume concentration measurements
- 8. Fluctuations in Molecule Numbers time large volume concentration small volume concentration time 1 molecule measurements
- 9. Influence of discreteness increases with decreasing volume! Fluctuations in Molecule Numbers time large volume concentration small volume concentration time 1 molecule measurements
- 10. Influence of discreteness increases with decreasing volume! How can we describe the observed stochasticity for small volumes? Fluctuations in Molecule Numbers time large volume concentration small volume concentration time 1 molecule measurements
- 11. discrete molecule number microscopic rate equations ni=k ni=k+1 r(n) chemical master equation (CME) nA=k nA=k+1 nA= j nB =k nA=j-1 nB=k+1 nA=i-1 nB=j-1 nC=k+1 nA=i nB=j nC =k ∅ A r0 A B r1 A + B C r2 1st order 2nd order 0th order Microscopic and Macroscopic Rate Equations elementaryreactionsstochastic
- 12. discrete molecule number microscopic rate equations ni=k ni=k+1 r(n) chemical master equation (CME) nA=k nA=k+1 nA= j nB =k nA=j-1 nB=k+1 nA=i-1 nB=j-1 nC=k+1 nA=i nB=j nC =k ∅ A r0 A B r1 A + B C r2 1st order 2nd order 0th order Microscopic and Macroscopic Rate Equations elementaryreactions high com putational com plexity high com putational com plexity stochastic
- 13. discrete molecule number microscopic rate equations ni=k ni=k+1 r(n) chemical master equation (CME) nA=k nA=k+1 nA= j nB =k nA=j-1 nB=k+1 nA=i-1 nB=j-1 nC=k+1 nA=i nB=j nC =k ∅ A r0 A B r1 A + B C r2 1st order 2nd order 0th order Microscopic and Macroscopic Rate Equations continuous concentration macroscopic rate equations averaging reaction rate equation (RRE) c=Sf(r,c,t) . fj=r0/Ω . fj=-r1cA . fj=-r2ΩcAcB... . elementaryreactions high com putational com plexity high com putational com plexity stochastic deterministic c=<n/Ω>
- 14. discrete molecule number microscopic rate equations ni=k ni=k+1 r(n) chemical master equation (CME) nA=k nA=k+1 nA= j nB =k nA=j-1 nB=k+1 nA=i-1 nB=j-1 nC=k+1 nA=i nB=j nC =k ∅ A r0 A B r1 A + B C r2 1st order 2nd order 0th order Microscopic and Macroscopic Rate Equations continuous concentration macroscopic rate equations averaging reaction rate equation (RRE) c=Sf(r,c,t) . fj=r0/Ω . fj=-r1cA . fj=-r2ΩcAcB... . elementaryreactions high com putational com plexity high com putational com plexity low com putational com plexity low com putational com plexity stochastic deterministic c=<n/Ω>
- 15. discrete molecule number microscopic rate equations ni=k ni=k+1 r(n) chemical master equation (CME) nA=k nA=k+1 nA= j nB =k nA=j-1 nB=k+1 nA=i-1 nB=j-1 nC=k+1 nA=i nB=j nC =k ∅ A r0 A B r1 A + B C r2 1st order 2nd order 0th order Microscopic and Macroscopic Rate Equations continuous concentration macroscopic rate equations averaging reaction rate equation (RRE) c=Sf(r,c,t) . fj=r0/Ω . fj=-r1cA . fj=-r2ΩcAcB... . elementaryreactions Is this description of the mean always accurate? high com putational com plexity high com putational com plexity low com putational com plexity low com putational com plexity stochastic deterministic c=<n/Ω>
- 16. Van Kampen’s System Size Expansion MEAN = RRE + VAR = LNA Ω-1 + Expansion of the moments of the CME in Ω -1 O(Ω-1) O(Ω-2) Linear Noise Approximation van Kampen (1981), Elsevier
- 17. Van Kampen’s System Size Expansion MEAN = RRE + VAR = LNA Ω-1 + Expansion of the moments of the CME in Ω -1 Exact for affine linear systems (only 0th and 1st order reactions) O(Ω-1) O(Ω-2) Linear Noise Approximation van Kampen (1981), Elsevier
- 18. Van Kampen’s System Size Expansion MEAN = RRE + VAR = LNA Ω-1 + Expansion of the moments of the CME in Ω -1 EMRE Ω-1 + O(Ω-2) Exact for affine linear systems (only 0th and 1st order reactions) Effective Mesoscopic Rate Equation O(Ω-2) Linear Noise Approximation van Kampen (1981), Elsevier
- 19. Van Kampen’s System Size Expansion MEAN = RRE + VAR = LNA Ω-1 + Expansion of the moments of the CME in Ω -1 EMRE Ω-1 + O(Ω-2) IOS Ω-2 + O(Ω-3) Exact for affine linear systems (only 0th and 1st order reactions) Effective Mesoscopic Rate Equation Inverse Omega Square Method Linear Noise Approximation van Kampen (1981), Elsevier Thomas et. al (2013), BMC Genomics
- 20. Van Kampen’s System Size Expansion MEAN = RRE + VAR = LNA Ω-1 + Expansion of the moments of the CME in Ω -1 EMRE Ω-1 + O(Ω-2) IOS Ω-2 + O(Ω-3) Exact for affine linear systems (only 0th and 1st order reactions) Effective Mesoscopic Rate Equation Inverse Omega Square Method Linear Noise Approximation van Kampen (1981), Elsevier Thomas et. al (2013), BMC Genomics
- 21. Van Kampen’s System Size Expansion MEAN = RRE + VAR = LNA Ω-1 + Expansion of the moments of the CME in Ω -1 EMRE Ω-1 + O(Ω-2) IOS Ω-2 + O(Ω-3) Exact for affine linear systems (only 0th and 1st order reactions) Effective Mesoscopic Rate Equation Inverse Omega Square Method How pronounced is this difference in approximation of mean/variance? Linear Noise Approximation van Kampen (1981), Elsevier Thomas et. al (2013), BMC Genomics
- 22. Van Kampen’s System Size Expansion MEAN = RRE + VAR = LNA Ω-1 + Expansion of the moments of the CME in Ω -1 EMRE Ω-1 + O(Ω-2) IOS Ω-2 + O(Ω-3) Exact for affine linear systems (only 0th and 1st order reactions) Effective Mesoscopic Rate Equation Inverse Omega Square Method How pronounced is this difference in approximation of mean/variance? Linear Noise Approximation van Kampen (1981), Elsevier Thomas et. al (2013), BMC Genomics
- 23. 0 0.05 0.1 0.15 0.2 0.25 Mean mRNA SSA EMRE RRE 0 0.1 0.2 0.3 0.4 Protein 0 0.2 0.4 0.6 0.8 1 Enzyme 0 10 20 300 1 2 3 4 5 x 10 −3 Time t Variance SSA IOS LNA 0 10 20 300 0.02 0.04 0.06 0.08 0.1 Time t 0 10 20 300 0.005 0.01 0.015 Time t Forward Simulation of System for Ω = 56 mRNA Protein Complex; kskdm k0 k1 km2 ; k2 ⌦ Enzyme
- 24. 0 0.05 0.1 0.15 0.2 0.25 Mean mRNA SSA EMRE RRE 0 0.1 0.2 0.3 0.4 Protein 0 0.2 0.4 0.6 0.8 1 Enzyme 0 10 20 300 1 2 3 4 5 x 10 −3 Time t Variance SSA IOS LNA 0 10 20 300 0.02 0.04 0.06 0.08 0.1 Time t 0 10 20 300 0.005 0.01 0.015 Time t Forward Simulation of System for Ω = 56 EMRE/IOS yield visibly better approximations to mean/variance!mRNA Protein Complex; kskdm k0 k1 km2 ; k2 ⌦ Enzyme
- 25. 0 0.05 0.1 0.15 0.2 0.25 Mean mRNA SSA EMRE RRE 0 0.1 0.2 0.3 0.4 Protein 0 0.2 0.4 0.6 0.8 1 Enzyme 0 10 20 300 1 2 3 4 5 x 10 −3 Time t Variance SSA IOS LNA 0 10 20 300 0.02 0.04 0.06 0.08 0.1 Time t 0 10 20 300 0.005 0.01 0.015 Time t Forward Simulation of System for Ω = 56 EMRE/IOS yield visibly better approximations to mean/variance! How does this enhanced approximation affect the inverse problem of parameter estimation? mRNA Protein Complex; kskdm k0 k1 km2 ; k2 ⌦ Enzyme
- 26. Formulation of the Inverse Problem Dataset: D = {tk, ¯y(tk), ik}nt k=1 Model: y(t; ✓) = h(x(t; ✓), ✓) with x sol. to ˙x = f(x, ✓)
- 27. Formulation of the Inverse Problem Dataset: D = {tk, ¯y(tk), ik}nt k=1 Model: y(t; ✓) = h(x(t; ✓), ✓) with x sol. to ˙x = f(x, ✓) Objective Function: JD(✓) = 1 2 ny X i=1 ntX k=1 ✓ ¯yi(tk) yi(tk; ✓) ik ◆2
- 28. Formulation of the Inverse Problem Dataset: D = {tk, ¯y(tk), ik}nt k=1 Model: y(t; ✓) = h(x(t; ✓), ✓) with x sol. to ˙x = f(x, ✓) Objective Function: JD(✓) = 1 2 ny X i=1 ntX k=1 ✓ ¯yi(tk) yi(tk; ✓) ik ◆2 ˆ✓D = arg min ✓ JD(✓)Fitted Parameter:
- 29. Formulation of the Inverse Problem Dataset: D = {tk, ¯y(tk), ik}nt k=1 Model: y(t; ✓) = h(x(t; ✓), ✓) with x sol. to ˙x = f(x, ✓) Objective Function: JD(✓) = 1 2 ny X i=1 ntX k=1 ✓ ¯yi(tk) yi(tk; ✓) ik ◆2 ˆ✓D = arg min ✓ JD(✓)Fitted Parameter: Hypothesis: Using higher orders of the system size expansion for mean/variance yield parameter estimates which are closer to the true parameter.
- 30. Workflow 50000 SSA samples generated with true paramer 1 dataset
- 31. Workflow 50000 SSA samples generated with true paramer 1 dataset subsampling 100 data ensembles
- 32. Workflow 50000 SSA samples generated with true paramer 1 dataset subsampling 100 data ensembles averaging 100 datasets D D D
- 33. Workflow 50000 SSA samples generated with true paramer 1 dataset subsampling 100 data ensembles global optimisation 100 parameters per dataset θ θ θ averaging 100 datasets D D D
- 34. Workflow 50000 SSA samples generated with true paramer 1 dataset subsampling 100 data ensembles global optimisation 100 parameters per dataset θ θ θ squared difference best estimated parameter vs. true parameter squared error squared error squared error averaging 100 datasets D D D
- 35. Workflow 50000 SSA samples generated with true paramer 1 dataset subsampling 100 data ensembles global optimisation 100 parameters per dataset θ θ θ squared difference best estimated parameter vs. true parameter squared error squared error squared error MSE = average over errors averaging 100 datasets D D D
- 36. Influence of Volume on Estimation Error (10000 samples/dataset) parameters involved in second order reaction mRNA Protein Complex; kskdm k0 k1 km2 ; k2 ⌦ Enzyme high error low error 10 −10 10 −5 10 0 log10(k2) 10 0 10 2 10 4 10 −10 10 −5 10 0 log10(k0) log10(kdm) 10 0 10 2 10 4 log10(ks) log10(k1) 10 0 10 2 10 4 log10(km2) 10 −12 10 −10 10 −8 10 −6 MSE log10(cE0) RRE EMRE 10 0 10 2 10 4 10 −10 10 −5 10 0 MSE log10( ) ΩΩ Ω Ω Ω
- 37. Influence of Volume on Estimation Error (10000 samples/dataset) parameters involved in second order reaction Error is volume-dependent and EMRE yields better parameter estimates than RRE! mRNA Protein Complex; kskdm k0 k1 km2 ; k2 ⌦ Enzyme high error low error 10 −10 10 −5 10 0 log10(k2) 10 0 10 2 10 4 10 −10 10 −5 10 0 log10(k0) log10(kdm) 10 0 10 2 10 4 log10(ks) log10(k1) 10 0 10 2 10 4 log10(km2) 10 −12 10 −10 10 −8 10 −6 MSE log10(cE0) RRE EMRE 10 0 10 2 10 4 10 −10 10 −5 10 0 MSE log10( ) ΩΩ Ω Ω Ω
- 38. Influence of Sample Size on Estimation Error samples: 10 100 1000 10000 #samples 10 0 10 2 10 4 10 0 10 2 10 4 10 −10 10 −5 10 0 MSE log10(k1) 10 0 10 2 10 4 10 0 10 2 10 4 ΩΩ Ω Ω
- 39. Influence of Sample Size on Estimation Error samples: 10 100 1000 10000 #samples Difference between EMRE and RRE increases with number of samples! 10 0 10 2 10 4 10 0 10 2 10 4 10 −10 10 −5 10 0 MSE log10(k1) 10 0 10 2 10 4 10 0 10 2 10 4 ΩΩ Ω Ω
- 40. 10 0 10 2 10 410 1 10 2 10 3 10 4 log10 (cE0) samples −1 0 1 2 3 4 log10 (k2 ) log10 (kdm ) log10 (k1 ) 10 0 10 2 10 4 log10 (k0 ) 10 0 10 2 10 4 log10 (ks ) 10 0 10 2 10 4 log10 (km2 ) 10 1 10 2 10 3 10 4 samples Ω Ω Ω Ω Influence of Sample Size and Volume on Estimation Error EMRE RRE log10(MSEEMRE/MSERRE)
- 41. 10 0 10 2 10 410 1 10 2 10 3 10 4 log10 (cE0) samples −1 0 1 2 3 4 log10 (k2 ) log10 (kdm ) log10 (k1 ) 10 0 10 2 10 4 log10 (k0 ) 10 0 10 2 10 4 log10 (ks ) 10 0 10 2 10 4 log10 (km2 ) 10 1 10 2 10 3 10 4 samples Ω Ω Ω Ω Influence of Sample Size and Volume on Estimation Error Difference between EMRE and RRE is largest in intermediate volume regime! EMRE RRE log10(MSEEMRE/MSERRE)
- 42. Model Comparision between RRE/EMRE 99% confidence interval 10 0 10 1 10 2 10 3 10 4 mean contribution to χ2 for EMRE Ω 10 0 10 1 10 2 10 3 10 4 10 1 10 2 10 3 10 4 mean contribution to χ2 for RRE Ω mean mRNA mean PROT mean ENZ TOTAL 0 0.020.04 pdfχ2 objective function at optimal point approximately ~ (#datapoints - #parameters) 2
- 43. Model Comparision between RRE/EMRE 99% confidence interval 10 0 10 1 10 2 10 3 10 4 mean contribution to χ2 for EMRE Ω 10 0 10 1 10 2 10 3 10 4 10 1 10 2 10 3 10 4 mean contribution to χ2 for RRE Ω mean mRNA mean PROT mean ENZ TOTAL 0 0.020.04 pdfχ2 objective function at optimal point approximately ~ (#datapoints - #parameters) 2 For low volumes EMRE model cannot be rejected based on chi2 test!
- 44. Model Comparision between LNA/IOS objective function at optimal point approximately ~ (#datapoints - #parameters) 2 10 0 10 1 10 2 10 3 10 4 mean contribution to χ2 for IOS Ω 10 0 10 1 10 2 10 3 10 4 mean contribution to χ2 for LNA Ω 0 0.010.02 pdfχ2 mean MRNA mean PROT mean ENZ var MRNA var PROT var ENZ TOTAL 10 1 10 2 10 3 10 4 10 5
- 45. Model Comparision between LNA/IOS objective function at optimal point approximately ~ (#datapoints - #parameters) IOS yields better approximations to variance than LNA 2 10 0 10 1 10 2 10 3 10 4 mean contribution to χ2 for IOS Ω 10 0 10 1 10 2 10 3 10 4 mean contribution to χ2 for LNA Ω 0 0.010.02 pdfχ2 mean MRNA mean PROT mean ENZ var MRNA var PROT var ENZ TOTAL 10 1 10 2 10 3 10 4 10 5
- 46. Conclusion and Outlook • EMRE can be used to obtain less biased parameter estimates •Advantage of EMRE increases with number of samples • Improvement in quality fit for EMRE vs RRE • Comparision to other methods such as moment equations •Apply to different models where information gain from variance is higher
- 47. Acknowledgements 2) Institute for Computational Biology Helmholtz Zentrum München Jan Hasenauer1,2 Fabian Theis1,2 Ramon Grima3 Philipp Thomas3 1) Chair for Mathematical Modeling of Biological Systems Technische Universität München 3) Centre for Synthetic and Systems Biology University of Edinburgh Thank you for your attention!