Transcript X - kitpc

Testing Generalized Einstein Relation
Ping Ao,
Beijing, August 11, 2016
Center for Systems Biomedicine, Shanghai Jiao Tong University
http://systemsbiology.sjtu.edu.cn
[email protected]
What is the Einstein relation (ER) ?
S D = 1 (Einstein, 1905)
detailed balance
What is the generalized Einstein relation (GER) ?
[S(x) + T(x)] D(x) [S(x) - T(x)] = S(x)
What is the fluctuation-dissipation theorem (FDT)?
Why is GER fundamental ?
On Uniqueness of "SDE Decomposition" in A-type Stochastic Integration
Yuan, Tang, Ao. Journal of Chemical Physics, 2016, accepted
http://120.52.73.78/arxiv.org/pdf/1603.07927v1.pdf
Can biology bring new fundamental law(s) to physics?
Erwin Schroedinger, What is Life? 1944
3rd Universal Dynamics in Nature
Biologically Motivated Fundamental Physics
My Biological involvements
•Genetic switch
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Robustness, Stability and Efficiency of Phage lambda Regulatory Network: Dynamical Structure Analysis,
Zhu, Yin, Hood, Ao. Journal of Bioinformatics and Computational Biology 2: 885-817 (2004);
Biological Sources of Intrinsic and Extrinsic Noise in cI Expression of Lysogenic Phage Lambda,
Lei, Tian, Zhu, Chen, Ao. Scientific Reports 5: 13597 (2015).
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One of three (3) well known quantitative biology problems 16 years ago.
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•Endogenous network theory on cancer genesis and progression
Cancer as Robust Intrinsic State of Endogenous Molecular-Cellular Network Shaped by Evolution.
Ao, Galas, Hood, Zhu, Medical Hypotheses 78: 678–684 (2008);
复杂疾病的系统医学视角：内源性网络理论
苏杭，王高伟，朱晓梅，徐岷涓，敖平. 自然杂志 37: 448-454 (2015)
•Metabolism
Towards Kinetic Modeling of Global Metabolic Networks with Incomplete Experimental Input on Kinetic Parameters,
Ao, Lee, Lidstrom, Yin, Zhu. Chinese J Biotechnology 24: 980-994 (2008);
Towards Stable Kinetics of Large Metabolic Networks: Nonequilibrium Potential Function Approach,
Chen, Yuan, Ao, Xu, Zhu. Phys. Rev. E93: 062409 (2016).
•Evolutionary dynamics
Laws in Darwinian Evolutionary Theory,
Ao. Physics of Life Reviews 2: 117-156 (2005);
Two-time-scale Population Evolution on a Singular Landscape,
Xu, Jiao, Jiang, Ao. Phys. Rev. E89: 012724 (2014).
Ao’s systems biology lab website:
http://systemsbiology.sjtu.edu.cn
•Statistical physics approach appears good in biology (e.g., this meeting), why?
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Theoretical questions
•Positive answer to Schrodinger’s question
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Word Equation of Darwin and Wallace (1858):
Evolution by Variation and Selection
“… it requires interpretation to bring out the originality
and force of the argument, more so than if Darwin
were a Newton or an Einstein abstracting far beyond
everyman’s power to follow and to understand. A hero
of science should be less accessible.”
C.C. Gillispie, The edge of objectivity, 1990
Key question: Can the word equation be quantified?
i.e., formulating evolutionary dynamics into a set of equations
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Laws in Darwinian Evolutionary Theory. P. Ao, Physics of Life Reviews 2 (2005) 116-156.
Meeting Challenge
• “… one of the biggest challenges of future research "
M. Pigliucci and J. Kaplan, Making Sense of Evolution: the conceptual
foundations of evolutionary biology (2006). pp203
• “A Major Scientific Problem in 21st Century”
David Gross, 2010, in a lecture at Shanghai Jiao Tong University, China
• Candidate
(Ao, 2005):
Word equation of Darwin and Wallace:
Evolution by Variation and Selection
→ dx/dt = f(x) + ζ(x, t),
< ζ > = 0, < ζ(x,t) (x,t’) > = 2 D(x,t) (t-t’)
• Two fundamental and quantitative concepts:
RA Fisher (1930): fundamental theorem of natural selection (fluctuation-dissipation theorem)
S Wright (1932): adaptive landscape
(energy-like function in stochastic processes)
Difficult but Important Problem
encountered in many fields
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Biology:
“ … the idea that there is such a quantity (adaptive landscape—P.A.) remains one of the most widely held
popular misconceptions about evolution”.
S.H. Rice, in Evolutionary Theory: mathematical and conceptual foundations (2004)
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Chemistry:
“The search for a generalized thermodynamic potential in the nonlinear range has attracted a great deal of
attention, but these efforts finally failed.”
G. Nicolis in New Physics, pp332 (1989)
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Physics:
“Statistical physicists have tried to find such a variational formulation for many years because, if it existed
in a useful form, it might be a powerful tool for the solution of many kinds of problems. My guess … is
that no such general principle exists.”
J. Langer in Critical Problems in Physics, pp26 (1997)
and, check recent issues of Physics Today, Physical Review Letters, …
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Mathematics:
gradient vs vector systems, unsolved (Holmes, 2006)
dissipative, f  0 ; asymmetric, f  0 (absence of detailed balance) ;
nonlinear ; stochastic with multiplicative noise
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Economy (econophysics), finance, engineering, …
Construction, I
• Standard stochastic differential equation:
dx/dt = f(x,t) + (x,t)
(S)
Gaussian and white, Wiener noise: <  > = 0, < (x,t)  (x,t’) > = 2 ε D (x,t) (t-t’)
• The desired equation (local and trajectory view):
[S(x,t) + T(x,t)] dx/dt = - ψ(x,t) +  (x,t)
(N)
Gaussian and white, Wiener noise: <  > = 0, < (x,t)   (x,t’) > = 2 ε S (x,t) (t-t’)
which has 4 dynamical elements:
dissipative, transverse, driving, and stochastic forces.
• The desired distribution (global and ensemble view):
steady state (Boltzmann-Gibbs) distribution eq(x) ~ exp(- ψ(x) / ε )
Construction, II
• Describing same phenomenon
[S(x,t) + T(x,t)] [f(x,t) + (x,t)] = - ψ(x,t) +  (x,t)
• Noise and deterministic “force” have independent origins
[S(x,t) + T(x,t)] f(x,t) = - ψ(x,t)
[S(x,t) + T(x,t)] (x,t) =  (x,t)
• Potential condition
 [S(x,t) + T(x,t)] f(x,t) = 0,
anti-symmetric, n(n-1)/2 equations
• Generalized Einstein-Sutherland relation
[S(x,t) + T(x,t)] D(x,t) [S(x,t) - T(x,t)] = S(x,t)
symmetric, n(n+1)/2 equations
• Total n2 conditions for the matrix [S(x,t) + T(x,t)] !
Hence, S, T, ψ can be constructed from D and f (Inverse is easy).
Ito, Stratonovich, and Present Method
on stochastic processes
1- d stochastic process
(Ao, Kwon, Qian, Complexity, 2007)
Stochastic differential equation (pre-equation according to van Kampen):
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dt q = f (q) + ζ(q, t),
Guassian-white noise, multiplicative
Ito Process:
• ∂tρ(q, t) = [ε∂q ∂q D(q) + ∂qD(q)φq(q) ] ρ(q, t)
ρ(q, t = ∞) ~ exp{ − φ (q) /ε } / D(q) ;
φ (q) = - ∫ dq f(q) / D(q)
Stratanovich process:
• ∂tρ(q, t) = [ε∂qD ½ (q)∂qD ½ (q) + ∂qD(q)φq(q) ] ρ(q, t)
ρ(q, t = ∞) ~ exp{ − φ (q) /ε } / D ½ (q)
Present process:
• ∂tρ(q, t) = [ ε∂qD(q)∂q + ∂qD(q)φq(q) ] ρ(q, t)
ρ(q, t = ∞) ~ exp{ − φ (q) /ε }
Gradient Expansion
Gradient expansion
Exact equations:
G ≡ (S + T ) -1
(Ao, 2004; 2005)
∂ × [G−1f(q)] = 0, G + Gτ = 2D.
=D+Q
Definition of F matrix from the “force” f in 2-d:
• F11 = ∂1f1, F12 = ∂2f1, F21 = ∂1f2, F22 = ∂2f2
Lowest order gradient expansion—linear matrix equation:
• GFτ − FGτ = 0
• G + Gτ = 2 D
Q = (FD − DFτ ) / tr(F)
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φ (q) = − ∫c dq · [G−1(q) f(q) ]
Stochastic dynamical structure analysis
(S(x.t) + T(x,t)) dx/dt = - grad ψ(x,t) +  (x,t)
(N)
Four dynamical elements, the representation:
S: Semi-positive definite symmetric matrix:
dissipation; degradation
T: Anti-symmetric matrix, transverse:
oscillation, conservative
ψ: Potential function:
capacity, cost function, landscape
: Stochastic force:
noise
Steady state distribution is determined by ψ(X),
when independent of time,
according to Boltzmann-Gibbs distribution, ρ ~ exp( - ψ / ε) !
Direct and quantitative measure for robustness and stability
Where we are
• A construction can be obtained generally
Ao, physics/0302081; Ao, J. Phys. A (2004); Yuan and Ao, J Stat Mech (2012)
• Linear dynamics,
stable or unstable
• Equivalence to other methods
Kwon, Ao, and Thouless, PNAS (2005)
Yin and Ao, J. Phys. A (2006);
• Limit cycles
Ao, Kwon, Qian, Complexity (2007)
Yuan et al. Chin Phys B (2013)
Zhu, Yin, Ao, Intl J. Mod. Phys. B (2007); Yuan et al, Phys Rev E (2013)
• Chaotic dynamics
• Applications:
Ma, Tan, Yuan, Yuan, Ao, Int’l J Bifurcation and Chaos (2014)
Phage lambda genetic switch:
Zhu et al, Funct Integr Genomics (2004)
Cancer genesis and progression:
Ao et al Med Hyp (2008); Yuan, Zhu, Radich, Ao. Scientific Reports (2016 )
• Gradient expansion: turned into a linear problem at each order
• Markov Processes:
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Ao, J. Phys. A (2004)
same three parts--energy function, diffusion, no-detailed balance
Ao, Chen, Shi. Chin Phys Lett (2013)
On Uniqueness of "SDE Decomposition" in A-type Stochastic Integration
Yuan, Tang, Ao. http://arxiv.org/abs/1603.07927 (2016)
New Predictions
• Generalized Einstein relation (Ao, J Phys A, 2004)
[S(x) + T(x)] D(x) [S(x) - T(x)] = S(x)
S D = 1 (Einstein, 1905)
T = 0, detailed balance
No: Luposchainsky and Hinrichsen, J Stat Phys, 2013
• Beyond Ito vs Stratonovitch (Ao, Kwon, Qian, 2007; Yuan, Ao, 2012)
No: Udo Seifert, Rep Prog Phys 2012
• “Free energy” equality
(Ao, Comm Theor Phys 2008; Tang et al, 2015)
< exp[ - ∫c dq · q(x, q) ] > = exp[ – ( F(end) – F(initial) )]
extension of Jarzynskii equality in physics
Open Problems
• Are those math curiosity or real physics?
Experimental efforts are needed!
• More stochastic equalities
generalized Einstein relation (not the fluctuation-dissipation relation)
• Quantum dynamics
Generalized Langevin equation
Lindblad equation
Acknowledgements
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Co-workers:
Ruoshi Yuan, Shanghai Jiao Tong Univ.
Ying Tang,
Shanghai Jiao Tong Univ.
Xinan Wang, Shanghai Jiao Tong Univ.
Shuyun Jiao, Shanghai Jiao Tong Univ.
Bo Yuan,
Shanghai Jiao Tong Univ.
Tianqi Chen, Shanghai Jiao Tong Univ.
Gaowei Wang, Shanghai Jiao Tong Univ.
Xue Lei,
Shanghai Jiao Tong Univ.
Lan Yin,
Peking Univ.
Xiaomei Zhu, GenMath, Seattle, USA
Yian Ma,
Univ Washington, Seattle, USA
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David Thouless, Physics, Univ. Washington, USA
Hong Qian,
Applied Math, Univ. Washington, USA
Chulan Kwon, Myongji Univ., S. Korea
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Funding:
USA: Institute for Systems Biology;
China: 985, 973
NIH
Thank you!