1. Noisy information transmission through molecularinteraction networksMichael P.H. Stumpf Theoretical Systems Biology Group09/05/2012Noise in Biological Networks Michael P.H. Stumpf 1 of 24

2. Cellular Decision Making Processes Noise in Biological Networks Michael P.H. Stumpf Cellular Decision Making Processes 2 of 24 3. What Do We Mean by a Cellular Decision? Cell Fates and Phenotypes How are decisions made? How non-random are thesedecisions? Why and how do cellular decisionDifferentiation Proliferation Apoptosis making processes diverge? Noise in Biological Networks Michael P.H. Stumpf Cellular Decision Making Processes 3 of 24 4. What Do We Mean by a Cellular Decision? Cell Fates and Phenotypes How are decisions made? How non-random are thesedecisions? Why and how do cellular decisionDifferentiation Proliferation Apoptosis making processes diverge? Information ProcessingEGF EGFRAkt S6 How reliable is the processing and transmission of information from the cells environment into the nucleus? Noise in Biological Networks Michael P.H. Stumpf Cellular Decision Making Processes 3 of 24 5. What Do We Mean by a Cellular Decision? Cell Fates and Phenotypes How are decisions made? How non-random are thesedecisions? Why and how do cellular decisionDifferentiation Proliferation Apoptosis making processes diverge? Information ProcessingEGF EGFRAkt S6 How reliable is the processing and transmission of information from the cells environment into the nucleus? Noise in Biological Networks Michael P.H. Stumpf Cellular Decision Making Processes 3 of 24 6. What Do We Mean by a Cellular Decision? Cell Fates and Phenotypes How are decisions made? How non-random are thesedecisions? Why and how do cellular decisionDifferentiation Proliferation Apoptosis making processes diverge? Information ProcessingEGF EGFRAkt S6 How reliable is the processing and transmission of information from the cells environment into the nucleus? Noise in Biological Networks Michael P.H. Stumpf Cellular Decision Making Processes 3 of 24 7. What Do We Mean by a Cellular Decision? Cell Fates and Phenotypes How are decisions made? How non-random are thesedecisions? Why and how do cellular decisionDifferentiation Proliferation Apoptosis making processes diverge? Information ProcessingEGF EGFRAkt S6 How reliable is the processing and transmission of information from the cells environment into the nucleus? Noise in Biological Networks Michael P.H. Stumpf Cellular Decision Making Processes 3 of 24 8. Signal Transduction vs. Signal Processing XX XY Y ZZ Z Noise in Biological Networks Michael P.H. Stumpf Information Transmission 4 of 24 9. Signal Transduction vs. Signal Processing Signal: S = {S1 , S2 , . . .} XX XY Y ZZ Z Noise in Biological Networks Michael P.H. Stumpf Information Transmission 4 of 24 10. Signal Transduction vs. Signal ProcessingCellular Information Signal: S = {S1 , S2 , . . .}Processing XX X We assume that X is areceptor and Z an effector (e.g.a transcription factor).We are interested in howY Y Z (t ) reects the stimulus pattern Si (t ). ZZ Z the molecular architecture Mj modulates the signal. Noise in Biological Networks Michael P.H. Stumpf Information Transmission 4 of 24 11. Signal Transduction vs. Signal ProcessingCellular Information Signal: S = {S1 , S2 , . . .}Processing XX X We assume that X is areceptor and Z an effector (e.g.a transcription factor).We are interested in howY Y Z (t ) reects the stimulus pattern Si (t ). ZZZ the molecular architecture Mj modulates the signal.S (t ) Z (t ) Z (t )ttt Noise in Biological Networks Michael P.H. Stumpf Information Transmission 4 of 24 12. Information TheoryNotions of Information (according to C. Shannon) Information is closely associated with uncertainty. What is signicant is the difculty in transmitting the message fromone point to another. Information in entropy: H (X ) = p(x ) log (p(x ))Noise in Biological Networks Michael P.H. Stumpf Information Transmission 5 of 24 13. Information TheoryNotions of Information (according to C. Shannon) Information is closely associated with uncertainty. What is signicant is the difculty in transmitting the message fromone point to another. Information in entropy: H (X ) = p(x ) log (p(x ))H (X , Y )H (X |Y )I (X , Y )H (Y |X )H (X )H (Y )Noise in Biological Networks Michael P.H. Stumpf Information Transmission5 of 24 14. Facets of InformationGaussian Channels TransmitterNoisy ChannelReceiverIf X is the message emanating from the transmitter and Y is themessage arriving at the receiver we model the channel as Pr(Y = y |X = x ) N(x , 2 ).C Noise in Biological Networks Michael P.H. Stumpf Information Transmission6 of 24 15. Facets of InformationGaussian Channels TransmitterNoisy Channel ReceiverIf X is the message emanating from the transmitter and Y is themessage arriving at the receiver we model the channel as Pr(Y = y |X = x ) N(x , 2 ).CThe mutual information I (Y , X ) captures the level of reduction inuncertainty about the state of Y once X is known.The channel capacity is the maximum information that can owthrough the channel,C = max I (Y , X ).p (x )This is a variational problem but straightforwardly solvable forGaussian and some other simple information channels. Noise in Biological Networks Michael P.H. StumpfInformation Transmission6 of 24 16. Information TransmissionThe Mutual Information between the signal, S, andthe effector, Z , is given by Pr(z , s)I (Z , S ) =Pr(z , s)S Pr(z )Pr(s)sS,z Z = H (Z ) H (Z |S ).XHigh mutual information means that Z faithfullyreproduces the signal S (as taken up by X ).ZNoise in Biological NetworksMichael P.H. Stumpf Information Transmission 7 of 24 17. Information TransmissionThe Mutual Information between the signal, S, andthe effector, Z , is given by Pr(z , s)I (Z , S ) =Pr(z , s)S Pr(z )Pr(s)SsS,z Z = H (Z ) H (Z |S ).XXHigh mutual information means that Z faithfullyreproduces the signal S (as taken up by X ).YSignal ProcessingUltimately, however, the signal is processed andaltered in a way not normally considered in e.g. theZZanalysis of Gaussian Channels.If we transduce a signal through a signallingnetwork/pathway it will get processed and distortedby intrinsic and extrinsic noise.Noise in Biological NetworksMichael P.H. Stumpf Information Transmission 7 of 24 18. Extrinsic vs. Intrinsic Noise S X Extrinsic Intrinsic Extrinsic + Intrinsic Z Noise in Biological Networks Michael P.H. Stumpf Information Transmission8 of 24 19. Extrinsic vs. Intrinsic Noise S X Extrinsic Intrinsic Extrinsic Y + Intrinsic Z Noise in Biological Networks Michael P.H. Stumpf Information Transmission8 of 24 20. Extrinsic vs. Intrinsic Noise S X Extrinsic Intrinsic Extrinsic Y + Intrinsic Z Noise in Biological Networks Michael P.H. Stumpf Information Transmission8 of 24 21. Some Strange ResultsSignal: S = {0, S } XX XY Y ZZ ZI (Z,X)0 .20 .1Noise in Biological Networks Michael P.H. Stumpf Information Transmission 9 of 24 22. Some Strange ResultsSignal: S = {0, S } XX XY Y ZZ ZI (Z,X) Stimulus On0 .20 .1Noise in Biological Networks Michael P.H. Stumpf Information Transmission 9 of 24 23. Some Strange ResultsXSignal: S = {0, S }X XX X X Y /Z Y /Z Y YX +Y ZZ ZZ ZI (Z,X) Stimulus On0 .20 .1Noise in Biological Networks Michael P.H. Stumpf Information Transmission 9 of 24 24. Information Propagation: Conclusions The dynamics of signal transduction cascades lead to dissipationof the signal, that is probably poorly described in terms ofGaussian channels. Under certain circumstances intrinsic and extrinsic noise canswamp or overwhelm the signal. Systematic differencesbetween parameters can even induce apparent correlationsbetween initial and terminal nodes in networks. Some of the results observable in stochastic treatments ofinformation propagation along signal-transduction networks appearcounter-intuitive.Noise in Biological Networks Michael P.H. Stumpf Information Transmission 10 of 24 25. Information Propagation: Conclusions The dynamics of signal transduction cascades lead to dissipationof the signal, that is probably poorly described in terms ofGaussian channels. Under certain circumstances intrinsic and extrinsic noise canswamp or overwhelm the signal. Systematic differencesbetween parameters can even induce apparent correlationsbetween initial and terminal nodes in networks. Some of the results observable in stochastic treatments ofinformation propagation along signal-transduction networks appearcounter-intuitive. Such results appear reect the interplay between the(deterministic) dynamics and the extrinsic and intrinsic sources ofnoise.Noise in Biological Networks Michael P.H. Stumpf Information Transmission 10 of 24 26. Information Propagation: Conclusions The dynamics of signal transduction cascades lead to dissipationof the signal, that is probably poorly described in terms ofGaussian channels. Under certain circumstances intrinsic and extrinsic noise canswamp or overwhelm the signal. Systematic differencesbetween parameters can even induce apparent correlationsbetween initial and terminal nodes in networks. Some of the results observable in stochastic treatments ofinformation propagation along signal-transduction networks appearcounter-intuitive. Such results appear reect the interplay between the(deterministic) dynamics and the extrinsic and intrinsic sources ofnoise. Rather than studying the propagation of information, we nextconsider the propagation of noise along a network.Noise in Biological Networks Michael P.H. Stumpf Information Transmission 10 of 24 27. Molecular Noise Noise in Biological Networks Michael P.H. Stumpf Modelling Molecular Noise 11 of 24 28. Molecular Noise Noise in Biological Networks Michael P.H. Stumpf Modelling Molecular Noise 11 of 24 29. Molecular Noise Noise in Biological Networks Michael P.H. Stumpf Modelling Molecular Noise 11 of 24 30. Molecular Noise Noise in Biological Networks Michael P.H. Stumpf Modelling Molecular Noise 11 of 24 31. Molecular Noise Noise in Biological Networks Michael P.H. Stumpf Modelling Molecular Noise 11 of 24 32. Molecular Noise Noise in Biological Networks Michael P.H. Stumpf Modelling Molecular Noise 11 of 24 33. Modelling Intrinsic NoiseThe stoichiometric matrix s11 212 ... s1r.... . . s21 ... S= . . ..... . . ...sm1... . . . smrdescribes the changes that can occuramong the r reactions for m molecularspecies. I.e. when reaction j occurs thenxi xi + sij . Komorowski et al., PNAS (2011); Komorowski et al., Bioinformatics (2012).Noise in Biological Networks Michael P.H. Stumpf Modelling Molecular Noise 12 of 24 34. Modelling Intrinsic NoiseThe stoichiometric matrix s11 212 ... s1rR1: m m + 1.... . . R2: m m 1 s21 ... S= . . R3: (m, p) (m, p + 1) ..... . . ... R4: p p 1sm1... . . . smr R5: (p, p ) (p 1, p + 1) describes the changes that can occur R6: (p, p ) (p + 1, p 1)among the r reactions for m molecular R7: p p 1species. I.e. when reaction j occurs thenxi xi + sij . Komorowski et al., PNAS (2011); Komorowski et al., Bioinformatics (2012).Noise in Biological Networks Michael P.H. Stumpf Modelling Molecular Noise 12 of 24 35. Modelling Intrinsic NoiseThe stoichiometric matrix s11 212 ... s1rR1: m m + 1.... . . R2: m m 1 s21 ... S= . . R3: (m, p) (m, p + 1) ..... . . ... R4: p p 1sm1... . . . smr R5: (p, p ) (p 1, p + 1) describes the changes that can occur R6: (p, p ) (p + 1, p 1)among the r reactions for m molecular R7: p p 1species. I.e. when reaction j occurs thenxi xi + sij . Intrinsic vs. Extrinsic Noise are due to stochastic effects because of small numbers of molecules and factors outside of the model, respectively. Extrinsic noise can, for example, affect the rates at which reactions occur. Komorowski et al., PNAS (2011); Komorowski et al., Bioinformatics (2012).Noise in Biological Networks Michael P.H. Stumpf Modelling Molecular Noise 12 of 24 36. Stochastic Dynamicsx (t ) = Noise in Biological Networks Michael P.H. Stumpf Modelling Molecular Noise 13 of 24 37. Stochastic Dynamicsx (t ) = x (0) Noise in Biological Networks Michael P.H. Stumpf Modelling Molecular Noise 13 of 24 38. Stochastic Dynamicsx (t ) = x (0) +SjBall et al., Ann. Appl. Probab. (2006).Sj Change in the state due to occurrence of jth reaction Noise in Biological Networks Michael P.H. Stumpf Modelling Molecular Noise13 of 24 39. Stochastic Dynamicstx (t ) = x (0) +Sj Yjfj (x , s)ds0Ball et al., Ann. Appl. Probab. (2006).Sj Change in the state due to occurrence of jth reactionYj (u ) is a Poisson point process (How many times did reaction j re?) Noise in Biological Networks Michael P.H. Stumpf Modelling Molecular Noise13 of 24 40. Stochastic Dynamicsr tx (t ) = x (0) + Sj Yj fj (x , s)dsj =10Ball et al., Ann. Appl. Probab. (2006).Sj Change in the state due to occurrence of jth reactionYj (u ) is a Poisson point process (How many times did reaction j re?) Noise in Biological Networks Michael P.H. Stumpf Modelling Molecular Noise13 of 24 41. Stochastic Dynamicsr tx (t ) = x (0) + Sj Yj fj (x , s)dsj =10Ball et al., Ann. Appl. Probab. (2006).Sj Change in the state due to occurrence of jth reactionYj (u ) is a Poisson point process (How many times did reaction j re?)Yj (u ) can be approximated as Y (u ) u + N(0, u ) Noise in Biological Networks Michael P.H. Stumpf Modelling Molecular Noise13 of 24 42. Stochastic Dynamicsr tx (t ) = x (0) +Sj Yjfj (x , s)dsj =10 Ball et al., Ann. Appl. Probab. (2006).Sj Change in the state due to occurrence of jth reactionYj (u ) is a Poisson point process (How many times did reaction j re?)Yj (u ) can be approximated as Y (u ) u + N(0, u )Deterministic approximationr t(t ) = (0) + Sjfj (, s)dsj =10 Noise in Biological Networks Michael P.H. Stumpf Modelling Molecular Noise 13 of 24 43. Decomposing Noise 1We want to know how much each reaction contributes to the overallnoise in the system and therefore dene x (t ) |jas the expectation of x (t ) conditioned on the processesY1 (t ), ..., Yj 1 (t ), Yj +1 (t ), ..., Yr (t ). It is a random variable wheretimings of reaction j have been averaged over all possible times,keeping all other reactions xed.Noise in Biological Networks Michael P.H. StumpfNoise Decomposition 14 of 24 44. Decomposing Noise 1We want to know how much each reaction contributes to the overallnoise in the system and therefore dene x (t ) |jas the expectation of x (t ) conditioned on the processesY1 (t ), ..., Yj 1 (t ), Yj +1 (t ), ..., Yr (t ). It is a random variable wheretimings of reaction j have been averaged over all possible times,keeping all other reactions xed.Thusx (t ) x (t ) |jis a random variable representing the difference between the nativeprocess x (t ) and a process with time-averaged jth reaction.Noise in Biological Networks Michael P.H. StumpfNoise Decomposition 14 of 24 45. Decomposing Noise 1We want to know how much each reaction contributes to the overallnoise in the system and therefore dene x (t ) |jas the expectation of x (t ) conditioned on the processesY1 (t ), ..., Yj 1 (t ), Yj +1 (t ), ..., Yr (t ). It is a random variable wheretimings of reaction j have been averaged over all possible times,keeping all other reactions xed.Thusx (t ) x (t ) |jis a random variable representing the difference between the nativeprocess x (t ) and a process with time-averaged jth reaction.Now the contribution of the jth reaction to the total variability of x (t ) is (j ) ( t ) ( x ( t ) x ( t )|j )(x (t ) x (t ) |j ) TNoise in Biological Networks Michael P.H. StumpfNoise Decomposition14 of 24 46. The Linear Noise Approximation (LNA) (t ) x (t ) (t ) =Noise in Biological Networks Michael P.H. Stumpf Noise Decomposition 15 of 24 47. The Linear Noise Approximation (LNA) (t ) x (t ) (t ) =rt t= x (0) (0) +Sjfj (x (s), s)ds fj ((s), s)ds j =100rt t+Sj Yjfj (x , s)ds fj (x (s), s)dsj =1 0 0Noise in Biological Networks Michael P.H. Stumpf Noise Decomposition15 of 24 48. The Linear Noise Approximation (LNA) (t ) x (t ) (t ) =rtt= x (0) (0) +Sjfj (x (s), s)ds fj ((s), s)ds j =10 0 t fj ((s), s)(s)ds0rt t+Sj Yjfj (x , s)ds fj (x (s), s)dsj =1 0 0Noise in Biological Networks Michael P.H. Stumpf Noise Decomposition 15 of 24 49. The Linear Noise Approximation (LNA) (t ) x (t ) (t ) =r tt= x (0) (0) +Sj fj (x (s), s)ds fj ((s), s)ds j =100t fj ((s), s)(s)ds 0rtt+Sj Yjfj (x , s)ds fj (x (s), s)dsj =1 00 t Wjfj ((s), s)ds 0Noise in Biological Networks Michael P.H. Stumpf Noise Decomposition15 of 24 50. The Linear Noise Approximation (LNA) (t ) x (t ) (t ) =r tt= x (0) (0) +Sj fj (x (s), s)ds fj ((s), s)ds j =10 0t fj ((s), s)(s)ds 0rtt+Sj Yjfj (x , s)ds fj (x (s), s)dsj =1 00 t Wjfj ((s), s)ds 0d = S F ()dt+ S diag F () dWNoise in Biological Networks Michael P.H. Stumpf Noise Decomposition15 of 24 51. The Linear Noise Approximation (LNA)In summary we have in the LNA:x (t ) = (t ) + (t )d = S F ()dt + S diagF () dW Noise in Biological Networks Michael P.H. Stumpf Noise Decomposition15 of 24 52. The Linear Noise Approximation (LNA)In summary we have in the LNA:x (t ) = (t ) + (t )d = S F ()dt + S diagF () dW The LNA implies a Gaussian distribution,x (t ) MVN ((t ), (t )) Noise in Biological Networks Michael P.H. Stumpf Noise Decomposition15 of 24 53. The Linear Noise Approximation (LNA)In summary we have in the LNA:x (t ) = (t ) + (t )d = S F ()dt + S diagF () dW The LNA implies a Gaussian distribution,x (t ) MVN ((t ), (t )) with mean (t ) given as s solution of the rate equation Noise in Biological Networks Michael P.H. Stumpf Noise Decomposition15 of 24 54. The Linear Noise Approximation (LNA)In summary we have in the LNA:x (t ) = (t ) + (t )d = S F ()dt + S diagF () dW The LNA implies a Gaussian distribution,x (t ) MVN ((t ), (t )) with mean (t ) given as s solution of the rate equation variances given as the solution ofd (t )= A(, t ) + A(, t )T + E (, t )E (, t )Tdt Noise in Biological Networks Michael P.H. Stumpf Noise Decomposition15 of 24 55. The Linear Noise Approximation (LNA)In summary we have in the LNA:x (t ) = (t ) + (t )d = S F ()dt + S diagF () dW The LNA implies a Gaussian distribution,x (t ) MVN ((t ), (t )) with mean (t ) given as s solution of the rate equation variances given as the solution ofd (t )= A(, t ) + A(, t )T + E (, t )E (, t )Tdt and covariances bycov(x (s), x (t )) = (s)(s, t )T for std (t , s) = A(, s)(t , s),(t , t ) = I . ds Noise in Biological Networks Michael P.H. Stumpf Noise Decomposition15 of 24 56. Decomposing Noise 2The LNA gives us much more exibility and allows us to write theoverall variance in x (t ), (t ), as the sum of the variances of individualreactions, (t ) = (1) (t ) + . . . + (r ) (t ).where we can obtain (t ) from d= A(t ) + A(t )T + D (t ) dtand the contribution to the overall variance resulting from reaction j asd (j ) = A(t )(j ) + (j ) A(t )T + D (j ) (t ).dt Noise in Biological Networks Michael P.H. Stumpf Noise Decomposition 16 of 24 57. Birth and death processWe consider the simple birth and death process,k+ xk x Noise in Biological Networks Michael P.H. StumpfNoise Decomposition 17 of 24 58. Birth and death processWe consider the simple birth and death process,k+ xk x How much of the noise comes from birth and how much comes fromdeath? Noise in Biological Networks Michael P.H. StumpfNoise Decomposition 17 of 24 59. Birth and death processWe consider the simple birth and death process,k+ xk x How much of the noise comes from birth and how much comes fromdeath?dx = (k + k x (t ))dt +k + dW1 +k x (t ) dW2birth noisedeath noise Noise in Biological Networks Michael P.H. Stumpf Noise Decomposition17 of 24 60. Birth and death processWe consider the simple birth and death process,k+ xk x How much of the noise comes from birth and how much comes fromdeath?dx = (k + k x (t ))dt +k + dW1 +k x (t ) dW2birth noisedeath noise We therefore have1 1=x+x2 2birth noisedeath noise Noise in Biological Networks Michael P.H. Stumpf Noise Decomposition17 of 24 61. Noise contributions - general resultsProposition 1In any open conversion system with only rst order reactions thecontribution of the product degradation to the variability in the productabundance is precisely one half of the total variability.1 (r ) = []nnnn2Komorowski M., Miekisz J., Stumpf M.P.H. Decomposing Noise in Biochemical Signalling Systems Highlights the Role of ProteinDegradation, Submitted, available on arXiv. Noise in Biological Networks Michael P.H. StumpfNoise Decomposition 18 of 24 62. Noise contributions - general resultsProposition 1In any open conversion system with only rst order reactions thecontribution of the product degradation to the variability in the productabundance is precisely one half of the total variability.1 (r ) = []nn nn 2Proposition 2In a general system the contribution of the product degradation to thevariability in the product abundance is one half its mean.1 (r )= xn nn 2Komorowski M., Miekisz J., Stumpf M.P.H. Decomposing Noise in Biochemical Signalling Systems Highlights the Role of ProteinDegradation, Submitted, available on arXiv. Noise in Biological Networks Michael P.H. StumpfNoise Decomposition 18 of 24 63. Linear cascades(A) X1 X2 X31 23456(B) X1 X2 X31 23456Noise in Biological Networks Michael P.H. Stumpf Noise Decomposition 19 of 24 64. Linear cascadesconversionslow conversionfast 7% 6% 36%(A)25% 50% 50% X1 X2 X31 234562% 1 3% 5% 2 8% 3% 3 4% catalyticslow 4catalytic fast(B)17%2% 520% 20% 6 X1 X2 X331%1 23456 10% 10% 33% 16%20% 20% 2%Noise in Biological Networks Michael P.H. Stumpf Noise Decomposition19 of 24 65. Linear cascadesconversionslow conversionfast 7% 6% 36%(A)25% 50% 50% X1 X2 X31 234562% 1 3% 5% 2 8% 3% 3 4% catalyticslow 4catalytic fast(B)17%2% 520% 20% 6 X1 X2 X331%1 23456 10% 10% 33% 16%20% 20% 2% Conversion reactions: Increasing the rates increases the contribution of earlier reactions to the overall noise. Catalytic reactions: Now also early degradation reactions start to contribute as the rates increase.Noise in Biological Networks Michael P.H. Stumpf Noise Decomposition19 of 24 66. Controlling Cell-to-Cell Variability Noise in Biological Networks Michael P.H. Stumpf Noise Decomposition 20 of 24 67. Controlling Cell-to-Cell VariabilityR1 R3R5 R6 R2R4 R7 Noise in Biological Networks Michael P.H. Stumpf Noise Decomposition 20 of 24 68. Controlling Cell-to-Cell VariabilityR1 R3R5 R6 R2R4 R7 127%347%5679%43%6% 28% < 1% No dephosphorylation.Noise in Biological Networks Michael P.H. Stumpf Noise Decomposition 20 of 24 69. Controlling Cell-to-Cell VariabilityR1 R3 R5R6R2 R4R7 1127% 7%23347%7%4556677 8%9%40%43%6%6% 3% 28% < 1% 30% No dephosphorylation. Slow dephosphorylation.Noise in Biological Networks Michael P.H. Stumpf Noise Decomposition20 of 24 70. Controlling Cell-to-Cell VariabilityR1 R3 R5R6R2 R4R7 1 1127% 7%2 5%23 3 5%347%7% 4424%5 556%6 667 77 8%9%5%40%43%6%6% 18%36% 3% 28% < 1% 30% No dephosphorylation. Slow dephosphorylation. Fast dephosphorylation.Noise in Biological Networks Michael P.H. Stumpf Noise Decomposition 20 of 24 71. Control of NoiseLet us consider the simple birth-death process with dx = (f (x ) g (x ))dt + f ( x )dW1 + g ( x )dW2 .birth noisedeath noise Noise in Biological Networks Michael P.H. StumpfNoise Decomposition21 of 24 72. Control of Noise Let us consider the simple birth-death process withdx = (f (x ) g (x ))dt + f ( x )dW1 + g ( x )dW2 . birth noisedeath noiseAt steady state we have12 g( x )(death) = . f ( x ) + gxx )x(For f (x ) = k , g (x ) = x this yields1(death) = 2 x .Noise in Biological Networks Michael P.H. StumpfNoise Decomposition21 of 24 73. Control of Noise Let us consider the simple birth-death process withdx = (f (x ) g (x ))dt + f ( x )dW1 + g ( x )dW2 . birth noisedeath noiseAt steady state we have12 g( x )(death) = . f ( x ) + gxx )x(For f (x ) = k , g (x ) = x this yields1(death) = 2 x .But if f (x ), g (x ) are Hill orMichaelis-Menten functions, thennoise can be reduced arbitrarily.Noise in Biological Networks Michael P.H. StumpfNoise Decomposition21 of 24 74. Control of Noise Let us consider the simple birth-death process withdx = (f (x ) g (x ))dt + f ( x )dW1 +g ( x )dW2 . birth noise death noiseAt steady state we haveControl of Degradation1.0 1.012 g( x )(death) = .0.8 0.8 f ( x ) + gxx )x( production and degradation rates normalised density0.6 0.6For f (x ) = k , g (x ) = x this yields1(death) = 2 x .0.4 0.4But if f (x ), g (x ) are Hill or0.2 0.2Michaelis-Menten functions, then0.0 0.0noise can be reduced arbitrarily. 0 1020 30 40xNoise in Biological Networks Michael P.H. StumpfNoise Decomposition 21 of 24 75. Control of NoiseNoise in Biological Networks Michael P.H. Stumpf Noise Decomposition 21 of 24 76. Noise Propagation: Conclusions We can numerically decompose the variance of stochasticchemical reaction networks and obtain easily interpretable anduseful insights. For a large class of cases we obtain exact analytical insights: For linear open conversion reactions we nd that output degradation contributes half the variance. For general systems the contribution of the product degradation reaction is half of the expected product abundance. Other measures of variability/dispersion can also be expressed inthis decomposition.Coefcient of Variation: 1 2 xFano Factor: 12 2 Noise can be controlled by regulating production and degradationreactions. Noise in Biological Networks Michael P.H. Stumpf Noise Decomposition 22 of 24 77. Open Questions (A Small Selection) How do non-linear dynamics and stochastic effects jointly affectthe dynamics and behaviour of biological systems? Is mutual information a good measure for the delity of information transmission? the efciency of biological information processing?Similar questions arise for other correlative measures (MIC etc ). Does the noise decomposition break down for small systems? extrinsic noise? non-linear dynamics? Can we manipulate noise through regulating the proteindegradation reaction? Do cells with controlled degradation of key-molecules behave in amore predictable manner? Can we develop stimulus patterns that increase or decreasecell-to-cell variability?Noise in Biological Networks Michael P.H. Stumpf Conclusions 23 of 24 78. Acknowledgements Noise in Biological Networks Michael P.H. Stumpf Conclusions 24 of 24