Cellular Dynamics From A Computational Chemistry PerspectiveHong QianDepartment of Applied MathematicsUniversity of Washington
The most important lesson learned from protein science is
The current state of affair of cell biology:
(1) Genomics: A,T,G,C symbols
(2) Biochemistry: molecules
Experimental molecular genetics defines the state(s) of a cell by their transcription pattern via expression level (i.e., RNA microarray).
Biochemistry defines the state(s) of a cell via concentrations of metabolites and copy numbers of proteins.
Protein Copy Numbers in Yeast Ghaemmaghami, S. et. al. (2003) Global analysis of protein expression in yeast, Nature, 425, 737-741.
Metabolites Levels in TomatoRoessner-Tunali et. al. (2003) Metabolic profiling of transgenic tomato plants , Plant Physiology, 133, 84-99.
But biologists define the state(s) of a cell by its phenotype(s)!
How does computational biology define the biological phenotype(s) of a cell in terms of the biochemical copy numbers of proteins?
The Chemical Master Equations: A New Mathematical Foundation of Chemical and Biochemical Reaction Systems
The Stochastic Nature of Chemical ReactionsSingle-molecule measurementsRelevance to cellular biology: small copy #Kramers theory for unimolecular reaction rate in terms of diffusion process (1940)Delbrcks theory of stochastic chemical reaction in terms of birth-death process (1940)
Single Channel Conductance
First Concentration Fluctuation Measurements (1972)
Fast Forward to 1998
Stochastic Enzyme Kinetics
Stochastic Chemistry (1940)
The Kramers theory and the CME clearly marked the domains of two areas of chemical research: (1) The computation of the rate constant of a chemical reaction based on the molecular structures, energy landscapes, and the solvent environment; and (2) the prediction of the dynamic behavior of a chemical reaction network, assuming that the rate constants are known for each and every reaction in the system.
Kramers Theory, Markov Process & Chemical Reaction Rate
But cellular biology has more to do with reaction systems and networks
Traditional theory for chemical reaction systems is based on the law of mass-action
Nonlinear Biochemical Reaction Systems and Kinetic Models
The Law of Mass Action and Differential Equations
Nonlinear Chemical Oscillations
A New Mathematical Theory of Chemical and Biochemical Reaction Systems based on Birth-Death Processes that Include Concentration Fluctuations and Applicable to small systems.
The Basic Markovian Assumption:The chemical reaction contain nX molecules of type X and nY molecules of type Y. X and Y bond to form Z. In a small time interval of Dt, any one particular unbonded X will react with any one particular unbonded Y with probability k1Dt + o(Dt), where k1 is the reaction rate.
A Markovian Chemical Birth-Death Process
Chemical Master Equation Formalism for Chemical Reaction SystemsM. Delbrck (1940) J. Chem. Phys. 8, 120.D.A. McQuarrie (1963) J. Chem. Phys. 38, 433.D.A. McQuarrie, Jachimowski, C.J. & M.E. Russell (1964) Biochem. 3, 1732.I.G. Darvey & P.J. Staff (1966) J. Chem. Phys. 44, 990; 45, 2145; 46, 2209. D.A. McQuarrie (1967) J. Appl. Prob. 4, 413. R. Hawkins & S.A. Rice (1971) J. Theoret. Biol. 30, 579.D. Gillespie (1976) J. Comp. Phys. 22, 403; (1977) J. Phys. Chem. 81, 2340.
Nonlinear Biochemical Reaction Systems: Stochastic Version
Stochastic Markovian Stepping Algorithm (Monte Carlo)l =q1+q2+q3+q4 = k1nA+ k-1n+ k2nB+ k3n(n-1)m
Picking Two Random Variables T & n derived from uniform r1 & r2 :fT(t) = l e -l t, T = - (1/l) ln (r1)Pn(m) = km/l , (m=1,2,,4) r2
Stochastic Oscillations: Rotational Random Walks
Defining Biochemical Noise
An analogy to an electronic circuit in a radioIf one uses a voltage meter to measure a node in the circuit, one would obtain a time varying voltage. Should this time-varying behavior be considered noise, or signal? If one is lucky and finds the signal being correlated with the audio broadcasting, one would conclude that the time varying voltage is in fact the signal, not noise. But what if there is no apparent correlation with the audio sound?
Continuous Diffusion Approximation of Discrete Random Walk Model
Stochastic Dynamics: Thermal Fluctuations vs. Temporal Complexity
Temporal dynamics should not be treated as noise!
A Second Example: Simple Nonlinear Biochemical Reaction System From Cell Signaling
We consider a simple phosphorylation-dephosphorylation cycle, or a GTPase cycle:
with a positive feedbackFerrell & Xiong, Chaos, 11, pp. 227-236 (2001)
Two ExamplesFrom Cooper and Qian (2008) Biochem., 47, 5681.From Zhu, Qian and Li (2009) PLoS ONE. Submitted
Simple Kinetic Model based on the Law of Mass Action
Bifurcations in PdPC with Linear and Nonlinear Feedbackc = 0c = 1c = 2
Markov Chain Representationv1w1v2w2v0w0
Steady State Distribution for Number Fluctuations
Large V Asymptotics
Beautiful, or Ugly Formulae
Bistability and Emergent Sates Pknumber of R* molecules: k
A Theorem of T. Kurtz (1971)In the limit of V , the stochastic solution to CME in volume V with initial condition XV(0), XV(t), approaches to x(t), the deterministic solution of the differential equations, based on the law of mass action, with initial condition x0.
We Prove a Theorem on the CME for Closed Chemical Reaction SystemsWe define closed chemical reaction systems via the chemical detailed balance. In its steady state, all fluxes are zero.For ODE with the law of mass action, it has a unique, globally attractive steady-state; the equilibrium state.For the CME, it has a multi-Poisson distribution subject to all the conservation relations.
Therefore, the stochastic CME model has superseded the deterministic law of mass action model. It is not an alternative; It is a more general theory.
The Theoretical Foundations of Chemical Dynamics and Mechanical MotionThe Semiclassical Theory.
Chemical basis of epi-genetics:
Exactly same environment setting and gene, different internal biochemical states (i.e., concentrations and fluxes). Could this be a chemical definition for epi-genetics inheritance?
Chemistry is inheritable!
Emergent Mesoscopic ComplexityIt is generally believed that when systems become large, stochasticity disappears and a deterministic dynamics rules.However, this simple example clearly shows that beyond the infinite-time in the deterministic dynamics, there is another, emerging stochastic, multi-state dynamics!This stochastic dynamics is completely non-obvious from the level of pair-wise, static, molecule interactions. It can only be understood from a mesoscopic, open driven chemical dynamic system perspective.
In a cartoon fashion(a)(b)(c)(d)
The mathematical analysis suggests three distinct time scales, and related mathematical descriptions, of (i) molecular signaling, (ii) biochemical network dynamics, and (iii) cellular evolution. The (i) and (iii) are stochastic while (ii) is deterministic.
The emergent cellular, stochastic evolutionary dynamics follows not gradual changes, but rather punctuated transitions between cellular attractors.
If one perturbs such a multi-attractor stochastic system:Rapid relaxation back to local minimum following deterministic dynamics (level ii);Stays at the equilibrium for a quite long tme;With sufficiently long waiting, exit to a next cellular state.
Relaxation, Wating, Barrier Crossing: R-W-BC of Stochastic Dynamics
In SummaryThere are two purposes of this talk: On the technical side, a suggestion on computational cell biology, and proposing the idea of three time scalesOn the philosophical side, some implications to epi-genetics, cancer biology and evolutionary biology.
Into the Future:Toward a Computational Elucidation of Cellular attractor(s) and inheritable epi-genetic phenotype(s)
What do We Need?It requires a theory for chemical reaction networks with small numbers of moleculesThe CME theory is an appropriate starting pointIt requires all the rate constants under the appropriate conditionsOne should treat the rate constants as the force field parameters in the computational macromolecular structures.
Analogue with Computational Protein Structures 40 yr agoWhile the equation is known in principle (Newtons equation), the large amount of unknown parameters (force field) makes a realistic computation essentially impossible.It has taken 40 years of continuous development to gradually converge to an acceptable set of parametersBut the issues are remarkably similar: defining biological (conformational) states, extracting the kinetics between them, and ultimately, functions.