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

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### 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

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

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 ?

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)

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

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

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)

S. Maslov, I. Ispolatov, PNAS, (2007);

Cascades of perturbations exponentially decay

(and signalternate) with network distance

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)

Perturbations – large changes of few proteins

- Fluctuations – small changes of many proteins

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

Image by Cell Signaling Technology, Inc: www.cellsignal.com

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)

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

Use false-positives in noisy high-throughput data!

18mM

log K(ns)

1M

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 1M (-14kT) detection thresholdof Y2H which is 3.6 std. dev. away

J. Zhang, SM, E. Shakhnovich, Molecular Systems Biology (2008)

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)

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

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 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);

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 Ci

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)

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

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 mexp[(E(ns)-E(s))/kT]

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)

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

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)

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

OK, protein binding networksare 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)

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

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)

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

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)

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

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 Ei > , 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

At equilibrium:

This leads to a set of nonlinear equations:

We can numerically solve these equations by iteration

Thus, given a set of total concentrations and a set of dissociation constants, equilibrium free and bound concentrations are uniquely determined.

Of course, the network is not always in equilibrium. There are fluctuations away from equilibrium:

Empirical PPI Network

PPI Net

Curated genome-wide network of PPI interactions in Baker’s Yeast (S. cerevisiae)

BIOGRID database:

Interactions independently confirmed in at least two published experiments

Protein abundance

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

and 77 multi-protein complexes

Dissociation constants

Dissociation constants are not presently empirically known

Minimum association necessary to bind a sizable fraction of dimers

Evolutionary motivated dissociation:

Denominator is chose to conform to the average association from the PINT database

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

Driven fluctuations

Driving fluctuations

Collective effects amplify fluctuations

Significant amplification (up to 20-fold) compared to isolated dimers

Is Collective Amplification Bound?

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:

where:

We can think of the set of total copy numbers as the size of the system

notation:

Suppressed concentration are unchanged

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