Mass-action equilibrium and non-specific interactions in protein interaction networks

1. Mass-action equilibrium and non-specific interactions in protein interaction networks Sergei Maslov Brookhaven National Laboratory

2. Living cells contain crowded and diverse molecular environments • Proteins constitute ~30% of E. coli and ~5% of yeast cytoplasm by weight • ~2000 protein types are • co-expressed • co-localized in yeast cytoplasm

3. Map of reproducible (>2 publications) protein-protein interactions in yeast If that’s not difficult enough:they are all interconnected • >80% of proteins are all connected in one giant cluster of PPI network • Small-world effect median network distance – 6 steps

4. Why small-world property might cause problems? • Interconnected binding networkscould indiscriminatelyspread perturbations • Systematic changes in expression: large changes in concentrations of a small number of proteins SM, I. Ispolatov, PNAS and NJP (2007) • Noise: small changes in concentrations of a large number of proteins K.-K. Yan, D. Walker, SM, PRL (2008) • How individual pathways can be turned on and off without upsetting the whole system ?

5. What about non-specific interactions? • Proteins form transient non-specific bonds with random, non-functional partners • For an organism to function specific interactions between proteins must dominate over non-specific ones: • How much stronger ~N specific interactions between N proteins need to be to overcome ~N2 non-specific interactions? • What limits it imposes on the number of protein typesand their concentrations? J. Zhang, SM, E. Shakhnovich, Molecular Systems Biology (2008)

6. My “spherical cow” assumptions • Protein concentrations Ci of all yeast proteins (under the rich growth medium conditions) and subcellular localizations are experimentally known (group of Weissman @ UCSF) • Consider only reproducible independently confirmed protein-protein interactions for non-catalytic binding (kinase-substrate pairs~5%) • The network: ~4000 heterodimers and ~100multi-protein complexes (we assume no cooperative binding in complexes) connecting ~1700 proteins • Know the relevant average of dissociation constants Kij~10nM. Turned out their distribution around this average DOES NOT MATTER MUCH!!! • Use “evolutionary motivated” binding strength: Kij=max(Ci, Cj)/const, which is sufficient to bind considerable fraction of twoproteins in a heterodimer

7. Law of Mass Action (LMA) • dDAB /dt= r (on)AB FA FB– r (off)AB DAB • In theequilibrium:DAB=FA FB /KAB;CA= FA+DAB ; CB= FB +DABor FA = CA /(1+ FB /KAB) and FB = CB /(1+ FA /KAB) • In a network:A system of ~2000 nonlinear equationsfor Fi that can be solved only numerically

8. Propagation of perturbations: the in silico study • Calculate the unperturbed (wildtype) LMA equilibrium • Simulate atwofold increaseof the concentration CA 2CA of just one type of proteinand recalculate equilibriumfree concentrations Fiof all other proteins • Look forcascading perturbations: A B  C  Dwith sign-alternation: A ( up), B ( down), C ( up), D ( down)

9. S. Maslov, I. Ispolatov, PNAS, (2007); Cascades of perturbations exponentially decay (and signalternate) with network distance

10. Mapping to resistor network • Conductivities ij– heterodimer concentrations Dij • Losses to the ground iG – free (unbound) concentrations Fi • Perturbations spread along linear chains loosely conducting to neighbors and ground • Mapping is exact for bi-partite networks  odd-length loops dampen perturbations S.Maslov, K. Sneppen, I. Ispolatov, New J. Phys, (2007)

14. Perturbations – large changes of few proteins • Fluctuations – small changes of many proteins

15. Two types of fluctuations in equilibrium concentrations • Driven fluctuations: changes in Dijdriven by stochastic variations in total concentrations Ci(random protein production/degradation) • Spontaneous fluctuations: stochastic changes in Dijat fixed Ci – described by equlibrium thermodynamics • Both types propagate through network <Dij2>network <Dij2>isolated

16. Image by Cell Signaling Technology, Inc: www.cellsignal.com Mitochondrial control of apoptosis

17. What limits do non-specific interactions impose on robust functioning of protein networks?J. Zhang, S. Maslov, E. Shakhnovich, MSB (2008) see talk on 8:48 AM in Room 411 (V39)

18. Competition between specific and nonspecific interactions • The effect of non-specific interactions grows with genome diversity m -- the number of co-expressed & co-localized proteins • Compare 3 equilibrium concentrations of a typical protein: • free (monomer) • specific heterodimer, • all non-specific heterodimers • Need to know: • protein concentrations: Ci • specific and non-specific dissociation constants:K(s)=K0exp(E(s)/kT), K(ns)=K0exp(E(ns)/kT

19. Ci Kij log(C/K0) “Evolutionary motivated”Kij=max(Ci, Cj)/10 1 nM 1 M

20. Use false-positives in noisy high-throughput data! 18mM log K(ns) 1M How to estimate E(ns)? • We estimate the median non-specific energy to beE(ns)=-4kT 2.5kT or K(ns)=18mM • Still thousands of pairs are below the 1M (-14kT) detection thresholdof Y2H which is 3.6 std. dev. away J. Zhang, SM, E. Shakhnovich, Molecular Systems Biology (2008)

21. mitochondria Phase diagram in yeast nucleus <C> • Evolution pushes the number of protein types mup for higher functional complexity, while keeping the concentration <C> is as low as possible to reduce the waste due to non-specific interactions • Still, on average proteins in yeast cytoplasm spend 20% of timebound in non-specific complexes cytoplasm J. Zhang, SM, E. Shakhnovich, Molecular Systems Biology (2008)

22. Collaborators and support • Koon-Kiu Yan, Dylan Walker, Tin Yau Pang (BNL/Stony Brook) • IaroslavIspolatov (Ariadne Genomics/BNL) • Kim Sneppen (Center for Models of Life, Niels Bohr Institute, Denmark) • Eugene Shakhnovich, Jingshan Zhang (Harvard) • DOE DMS DE-AC02-98CH10886 • NIH/NIGMS R01 GM068954

23. Thank you!

24. Conclusions • Time to go beyond topology of PPI networks! • Interconnected networks present a challenge for robustness: • Perturbations and noise • Non-specific interactions • We were the first to attempt quantifying these effects on genome-wide scale • Estimates will get better as we get better data on kinetic & equilibrium constants

25. Collaborators, papers, and support • Koon-Kiu Yan, Dylan Walker, Tin Yau Pang (BNL/Stony Brook) • Iaroslav Ispolatov (Ariadne Genomics/BNL) • Kim Sneppen (Center for Models of Life, Niels Bohr Institute, Denmark) • Eugene Shakhnovich, Jingshan Zhang (Harvard) • DOE Division of Material Science, DE-AC02-98CH10886 • NIH/NIGMS, R01 GM068954 Propagation of large concentration changes in reversible protein binding networks, S. Maslov, I. Ispolatov, PNAS 104:13655 (2007); Constraints imposed by non-functional protein–protein interactions on gene expression and proteome size, J. Zhang, S. Maslov, E. Shakhnovich, Molecular Systems Biology 4:210 (2008); Fluctuations in Mass-Action Equilibrium of Protein Binding NetworksK-K. Yan, D. Walker, S. Maslov, Phys Rev. Lett., 101, 268102 (2008); Spreading out of perturbations in reversible reaction networksS. Maslov, K. Sneppen, I. Ispolatov, New Journal of Physics 9: 273 (2007); Topological and dynamical properties of protein interaction networks. S. Maslov, book chapter in the " Protein-protein interactions and networks: Identification, Analysis and Prediction“, Springer-Verlag (2008);

26. Collective Effects Amplify Spontaneous Noise Collective effects significantly amplify (up to a factor of 20) spontaneous noise Is there an upper bound to this amplification?

27. Stochastic fluctuations in D*ij at fixed Ci Free energy G, for a given occupation state Here is not independent but related to via

28. What limits do non-specific interactions impose on robust functioning of protein networks?J. Zhang, S. Maslov, E. Shakhnovich, Molecular Systems Biology (2008)

29. Competition between specific and nonspecific interactions • The effect of non-specific interactions grows with m -- the number of co-expressed & co-localized proteins • Assume a protein is biologically active when bound to its uniquespecific interaction partner • Compare 3 equilibrium concentrations:free (monomer),specific dimer,all non-specific dimers • Need to know the average and distributions of: • protein concentrations: C • specific and non-specific dissociation constants:K(s)=K0exp(E(s)/kT), K(ns)=K0exp(E(ns)/kT) • Dimensionless parameters: log(C/K0),E(s)/kT, E(ns)/kT

30. C m Limits on parameters • For specific dimers to dominate over monomers: C  K(s)= =K0exp(E(s)/kT) • For specific interactions to dominate over non-specific: C/K(s) mC/K(ns)or mexp[(E(ns)-E(s))/kT]

32. Intra-cellular noise • Noise typically means fluctuations in total concentrations Ci (e.g. cell-to-cell variability measured for of all yeast proteins by Weissman lab @ UCSF) • Needs to be converted into noise in biologically relevantdimer (Dij) or monomer (Fi) concentrations • Two types of noise: intrinsic (uncorrelated) and extrinsic (correlated) (M. Elowitz, U. Alon, et. al. (2005)) • Intrinsic noise could be amplified by the conversion (sometimes as much as 30 times!) • Extrinsic noise partially cancels each other • Essential proteins seem to be more protected from noiseand perturbations PNAS (2007), Phys. Rev. Lett. (2008)

33. Going beyond topology • We already know a lot about topology of complex networks (scale-free, small-world, clustering, etc) • Network is just a backbone for complex dynamical processes • Time to put numbers on nodes/edges and study these processes • For binding networks – governed by law of mass action

34. D The total number of cascades is still significant • The fraction of significantly (> noise level ~ 20%) affected proteins at distance D quickly decays --> exp(-  D) • The total number of neighbors at distance D quickly rises --> exp( D) • The number of affected proteins at distance D slowly decays--> exp(- (- )D) SM, I. Ispolatov, PNAS (2007)

35. Free concentrations: Fi Bound concentrations: Dij Spearman rank correlation: 0.89 Pearson linear correlation: 0.98 Spearman rank correlation: 0.89 Pearson linear correlation: 0.997 Robustness with respect to assignment of Kij SM, I. Ispolatov, PNAS, 104,13655-13660 (2007)

36. OK, protein binding networksare robust, but can cascading changes be used to send signals?

37. Robustness: Cascades of perturbations on average exponentially decay S.Maslov, K. Sneppen, I. Ispolatov, NJP (2007)

38. More robust Less robust How robust is the mass-action equilibrium against perturbations?

39. HHT1 SM, I. Ispolatov, PNAS, 104,13655-13660 (2007)

40. SM, I. Ispolatov, PNAS, 104,13655-13660 (2007)

41. SM, I. Ispolatov, PNAS, 104,13655-13660 (2007)

42. Perturbations propagate along dimers with large concentrations • They cascade down theconcentration gradient and thus directional • Free concentrations of intermediate proteins are low SM, I. Ispolatov, PNAS, 104,13655-13660 (2007)

43. Non-specific phase diagram

44. Three states of a protein • Each protein i has 3 possible states: Ci=[ii’]+[i]+[iR] • Concentrations are related by the Law of Mass Action • Compare the 3 concentrations: [ii’] should dominate

45. Non-specific binding energies

46. Model of nonspecific interactions • Assume for nonspecific interactions scales with sum of surface hydrophobicities of two proteins • Distribution of fraction of hydrophobicAas on protein’s surface • Distribution of is Gaussian (proportional to hydrophobicity) E. J. Deeds, O. Ashenberg, and E. I. Shakhnovich, PNAS 103, 311 (2006)

47. Parameters of non-specific interactions out of high-throughput Y2H experiments • Detection threshold Kd* of Kijin Yeast 2-Hybrid experiments J. Estojak, R. Brent and E. A. Golemis. Mol. Cell. Biol.15, 5820 (1995) • Interaction detected in Y2H if < E* • If  pairwise interactions are detected among N protein types

48. Chemical potential description of non-specific interactions between proteins

49. Chemical potential of the system • More hydrophobic surface  more likely to bind nonspecific. Probability to be monomeric follows the Fermi-Dirac distribution • [i]>[iR] for Ei >  , and vise versa • Find the chemical potential by solving