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

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

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 Map of reproducible (>2 publications) protein-protein interactions in yeast

Why small-world property might cause problems? Interconnected binding networks could indiscriminately spread 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 ?

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 ~N 2 non-specific interactions? What limits it imposes on the number of protein types and their concentrations? J. Zhang, SM, E. Shakhnovich, Molecular Systems Biology (2008)

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

Law of Mass Action (LMA) dD AB /dt = r (on) AB F A F B – r (off) AB D AB In the equilibrium: D AB =F A F B /K AB ; C A = F A +D AB ; C B = F B +D AB or F A = C A /(1+ F B /K AB ) and F B = C B /(1+ F A /K AB ) In a network: A system of ~2000 nonlinear equations for F i that can be solved only numerically

Propagation of perturbations: the in silico study Calculate the unperturbed (wildtype) LMA equilibrium Simulate a twofold increase of the concentration C A  2C A of just one type of protein and recalculate equilibrium free concentrations F i of all other proteins Look for cascading perturbations: A  B  C  D with sign-alternation: A (  up), B (  down), C (  up), D (  down)

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

Mapping to resistor network Conductivities  ij – heterodimer concentrations D ij Losses to the ground  iG – free (unbound) concentrations F i 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)

Perturbations – large changes of few proteins Fluctuations – small changes of many proteins

Two types of fluctuations in equilibrium concentrations Driven fluctuations: changes in D ij driven by stochastic variations in total concentrations C i (random protein production/degradation) Spontaneous fluctuations: stochastic changes in D ij at fixed C i – described by equlibrium thermodynamics Both types propagate through network  network  isolated

Image by Cell Signaling Technology, Inc: Mitochondrial control of apoptosis

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)

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: C i specific and non-specific dissociation constants: K (s) =K 0 exp(E (s) /kT), K (ns) =K 0 exp(E (ns) /kT Competition between specific and nonspecific interactions

log(C/K 0 ) K ij “Evolutionary motivated” K ij =max(C i, C j )/10 1  M 1 nM C i

We estimate the median non-specific energy to be E (ns) =-4kT  2.5kT or K (ns) =18mM Still thousands of pairs are below the 1  M (-14kT) detection threshold of Y2H which is 3.6 std. dev. away J. Zhang, SM, E. Shakhnovich, Molecular Systems Biology (2008) How to estimate E (ns) ? 1M1M 18mM log K (ns) Use false-positives in noisy high-throughput data!

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

Collaborators 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 DMS DE-AC02-98CH10886 NIH/NIGMS R01 GM068954

Thank you!

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

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 GM Propagation of large concentration changes in reversible protein binding networks, S. Maslov, I. Ispolatov, PNAS 104:13655 (2007); 2.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); 3.Fluctuations in Mass-Action Equilibrium of Protein Binding Networks K-K. Yan, D. Walker, S. Maslov, Phys Rev. Lett., 101, (2008); 4.Spreading out of perturbations in reversible reaction networks S. Maslov, K. Sneppen, I. Ispolatov, New Journal of Physics 9: 273 (2007); 5.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);

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?

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

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

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 unique specific 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) =K 0 exp(E (s) /kT), K (ns) =K 0 exp(E (ns) /kT) Dimensionless parameters: log(C/K 0 ), E (s) /kT, E (ns) /kT Competition between specific and nonspecific interactions

Limits on parameters For specific dimers to dominate over monomers: C  K (s)= =K 0 exp(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] m C

Intra-cellular noise Noise typically means fluctuations in total concentrations C i (e.g. cell-to-cell variability measured for of all yeast proteins by Weissman UCSF) Needs to be converted into noise in biologically relevant dimer (D ij ) or monomer (F i ) 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 noise and perturbations PNAS (2007), Phys. Rev. Lett. (2008)

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

SM, I. Ispolatov, PNAS (2007) 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) D

Robustness with respect to assignment of K ij Spearman rank correlation: 0.89 Pearson linear correlation: 0.98 Bound concentrations: D ij Spearman rank correlation: 0.89 Pearson linear correlation: Free concentrations: F i SM, I. Ispolatov, PNAS, 104, (2007)

OK, protein binding networks are robust, but can cascading changes be used to send signals?

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

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

SM, I. Ispolatov, PNAS, 104, (2007) HHT1

SM, I. Ispolatov, PNAS, 104, (2007)

Perturbations propagate along dimers with large concentrations They cascade down the concentration gradient and thus directional Free concentrations of intermediate proteins are low SM, I. Ispolatov, PNAS, 104, (2007)

Non-specific phase diagram

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

Non-specific binding energies

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

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

Chemical potential description of non-specific interactions between proteins

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 E i > , and vise versa Find the chemical potential  by solving

Network Equilibrium Given a set of total concentrations and the protein interaction network, we can determine the equilibrium bound and unbound concentrations We can numerically solve these equations by iteration At equilibrium: This leads to a set of nonlinear equations: Of course, the network is not always in equilibrium. There are fluctuations away from equilibrium: Thus, given a set of total concentrations and a set of dissociation constants, equilibrium free and bound concentrations are uniquely determined.

Empirical PPI Network Curated genome-wide network of PPI interactions in Baker’s Yeast (S. cerevisiae) BIOGRID database: Interactions independently confirmed in at least two published experiments Genome wide set of protein abundances during log-phase growth Retain only interactions between proteins of known total concentration 1740 proteins involved in 4085 heterodimers PPI Net Protein abundance Dissociation constants Dissociation constants are not presently empirically known Denominator is chose to conform to the average association from the PINT database Evolutionary motivated dissociation: Minimum association necessary to bind a sizable fraction of dimers and 77 multi-protein complexes

Driven Fluctuations Consider a set of total concentrations that are typical in the cell (i.e., when the cell is in log growth phase) We want to examine small deviations in total concentration that arise as a result of: 1) Upstream noise in genetic regulation 2) Stochastic fluctuations in protein production/degradation mechanisms The typical time-scale of these small fluctuations in total concentration is minutes. We refer to these as driving fluctuations because they propagate through the network and drive fluctuations in dimer concentration network Driving fluctuations Driven fluctuations

Collective effects amplify fluctuations Significant amplification (up to 20-fold) compared to isolated dimers

Is Collective Amplification Bound? where: To answer this question, let us calculate the noise from the partition function using an alternate formalism Calculate average statistical quantities in the usual way: We can think of the set of total copy numbers as the size of the system notation: Suppressed concentration are unchanged