Mass-action equilibrium and non-specific interactions in

protein interaction networks

Sergei MaslovBrookhaven 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 ~N2 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 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 ~100 multi-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

102

103

104

105

10610

0

101

102

103

104

protein abundance (copies/cell)hi

stog

ram

Law of Mass Action (LMA)

dDAB /dt = r (on)AB FA FB – r (off)

AB DAB

In the equilibrium:

DAB=FA FB /KAB ; CA= FA+DAB ; CB= FB +DAB

or 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 .

1 /i

ij ij

j nn i

CF

F K

Propagation of perturbations: the in silico study

Calculate the unperturbed (wildtype) LMA equilibrium

Simulate a twofold increase of the concentration CA 2CA of just one type of protein and recalculate equilibrium free concentrations Fi 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 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 Dij driven by stochastic variations in total concentrations Ci (random protein production/degradation)

• Spontaneous fluctuations: stochastic changes in Dij at fixed Ci – described by equlibrium thermodynamics

• Both types propagate through network <Dij

2>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)

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

Competition between specific and nonspecific interactions

log(C/K0)

Kij

“Evolutionary motivated” Kij=max(Ci, Cj)/10

1 M1 nM

Ci

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 threshold of Y2H which is 3.6 std. dev. away

literature Species Fraction

(Ito et al. , 2001) Yeast 4549 3.6 (Li et al. , 2004) C. elegans 4027 1873 10000 3.6

(Giot et al. , 2003) Drosophila 20439 11282 10306 3.5(Stelzl et al. , 2005) Human 3186 4456 5632 3.6(Rual et al. , 2005) Human 2754 3.87200

6000

# s t dbaitN preyN

41053.2 41015.2 41052.3 41054.2 41006.1

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

How to estimate E(ns)?

1M

42 1018mM

log K(ns)

Use false-positives in noisy high-throughput data!

<C>

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 <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 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 GM068954

1. 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 NetworksK-K. Yan, D. Walker, S. Maslov, Phys Rev. Lett., 101, 268102 (2008);

4. Spreading out of perturbations in reversible reaction networksS. 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 Ci

*

*

*

1

*

( )

*

log( / ) log(

log( / )

/ ) N

B

B ij ijiij

i i i i

i

k

j j

k T F F

k TD D

e

e

e

D

C C

G

Free energy G, for a given occupation state }{ *ijD

Here is not independent but related to via * *i i im

m i

F C D

*iF

*imD

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)=K0exp(E(s)/kT), K(ns)=K0exp(E(ns)/kT) Dimensionless parameters: log(C/K0), 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)= =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]

m

C

( )

*

**

1

* *

* *log( / )]

log( / ), where

[

i i i i

ij Eij ij

ij

N

Bi j

ij ijBG

k T

k D D

F F e

e

C

D T

F D

ò

ò

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 relevant dimer (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 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 Kij

Spearman rank correlation: 0.89

Pearson linear correlation: 0.98

Bound concentrations: Dij

Spearman rank correlation: 0.89

Pearson linear correlation: 0.997

Free concentrations: Fi

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

1 2 3 4 5 610

-14

10-12

10-10

10-8

10-6

10-4

10-2

100

distance from perturbation source

aver

age

rela

tive

chan

ge in

Fi

1nM10nM0.1M1M10M0.1mM

How robust is the mass-action equilibrium against perturbations?

Less robust

More robust

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

HHT1

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

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

Perturbations propagate along dimers with large concentrations

They cascade down the concentration gradient and thus directional

Free concentrations of intermediate proteins are lowSM, I. Ispolatov, PNAS, 104,13655-13660 (2007)

Non-specific phase diagram

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

Non-specific binding energies

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)

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 Kd* of Kij 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 Ei > , and vise versa

Find the chemical potential by solving

/ ( ) /[ ] /[ ] i B i B

E k T E k Ti iR const e e

( ) /

( ) /10

[ ] 1( )

[ ] 1

i j

i

j

E EE kT R kT

E kTj j

CiRe e f E dE

i C e

100

101

102

103

104

105

10610

0

101

102

103

104

protein abundance (copies/cell)

hist

ogra

m

1000 E. coli proteins3868 yeast proteins

Network Equilibrium

Given a set of total concentrations and the protein interaction network, we can determine the equilibrium bound and unbound concentrations

jiijj

ii KF

CF

/1We can numerically solve these equations by iteration

ji ij

jiii K

FFFCAt equilibrium:

This leads to a set of nonlinear equations:

ii CF )0(

jiij

nj

ini

KF

CF

/1 )(

)1(

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

iii FFF * ijijij DDD *

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:20

),max( jiij

CCK

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

iii CCC *

iC

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

networkiC ijD

Driving fluctuations

Driven fluctuations

Collective effects amplify fluctuations

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

Is Collective Amplification Bound?

ijij

ijij

ijij

ij

ij

ijij

ij

ij

DD

DD

DD

D

D

DD

D

D

1)1(

)()()(

**

2*2*2 where:

ijij DD *

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: iC

x

)}{,()( ijjii CCZCZ We can think of the set of total copy numbers as the size of the system

notation:

Suppressed concentration are unchanged