Basic Enzyme Model Futile Cycle (Single Equilibrium) 2 -Site Phosphorylation Chain (Multiple Equilibria) Math 609: Mathematical Methods for Systems Biology Guest Lecture Matthew Douglas Johnston Van Vleck Visiting Assistant Professor University of Wisconsin-Madison Tuesday, May 6, 2014 Matthew Douglas Johnston Guest Lecture
Basic Enzyme Model Futile Cycle (Single Equilibrium) 2 -Site Phosphorylation Chain (Multiple Equilibria) 1 Basic Enzyme Model Set-up Properties Numerical Simulation Matthew Douglas Johnston Guest Lecture
Basic Enzyme Model Futile Cycle (Single Equilibrium) 2 -Site Phosphorylation Chain (Multiple Equilibria) 1 Basic Enzyme Model Set-up Properties Numerical Simulation 2 Futile Cycle (Single Equilibrium) Set-up Properties Numerical Simulation Matthew Douglas Johnston Guest Lecture
Basic Enzyme Model Futile Cycle (Single Equilibrium) 2 -Site Phosphorylation Chain (Multiple Equilibria) 1 Basic Enzyme Model Set-up Properties Numerical Simulation 2 Futile Cycle (Single Equilibrium) Set-up Properties Numerical Simulation 3 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation Matthew Douglas Johnston Guest Lecture
Basic Enzyme Model Set-up Futile Cycle (Single Equilibrium) Properties 2 -Site Phosphorylation Chain (Multiple Equilibria) Numerical Simulation 1 Basic Enzyme Model Set-up Properties Numerical Simulation 2 Futile Cycle (Single Equilibrium) Set-up Properties Numerical Simulation 3 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation Matthew Douglas Johnston Guest Lecture
Basic Enzyme Model Set-up Futile Cycle (Single Equilibrium) Properties 2 -Site Phosphorylation Chain (Multiple Equilibria) Numerical Simulation Basic Michaelis-Menten Enzyme Model is k + 1 k 2 S + E → P + E ⇄ C k − 1 where 1 S is a substrate (e.g. unphosphorylated protein) 2 E is an enzyme 3 C is an intermediate compound (really, C = SE ) 4 P is a product (e.g. phosphorylated protein) 5 k + 1 , k − 1 , and k 2 are (positive) rate constants Matthew Douglas Johnston Guest Lecture
Basic Enzyme Model Set-up Futile Cycle (Single Equilibrium) Properties 2 -Site Phosphorylation Chain (Multiple Equilibria) Numerical Simulation Dynamics ( mass-action model ) given by: s = − k + ˙ 1 s · e + k − 1 c e = − k + ˙ 1 s · e + ( k − 1 + k 2 ) c c = k + ˙ 1 s · e − ( k − 1 + k 2 ) c p = k 2 c ˙ where s = [ S ], e = [ E ], c = [ C ], and p = [ P ]. Matthew Douglas Johnston Guest Lecture
Basic Enzyme Model Set-up Futile Cycle (Single Equilibrium) Properties 2 -Site Phosphorylation Chain (Multiple Equilibria) Numerical Simulation Dynamics ( mass-action model ) given by: s = − k + ˙ 1 s · e + k − 1 c e = − k + ˙ 1 s · e + ( k − 1 + k 2 ) c c = k + ˙ 1 s · e − ( k − 1 + k 2 ) c p = k 2 c ˙ where s = [ S ], e = [ E ], c = [ C ], and p = [ P ]. Distressing observation : system is 4-dimensional and has undetermined parameters. :-( Matthew Douglas Johnston Guest Lecture
Basic Enzyme Model Set-up Futile Cycle (Single Equilibrium) Properties 2 -Site Phosphorylation Chain (Multiple Equilibria) Numerical Simulation What properties can we use to analyse this model? Matthew Douglas Johnston Guest Lecture
Basic Enzyme Model Set-up Futile Cycle (Single Equilibrium) Properties 2 -Site Phosphorylation Chain (Multiple Equilibria) Numerical Simulation What properties can we use to analyse this model? There are two conservation laws : s + ˙ ˙ c + ˙ p = 0 = s ( t ) + c ( t ) + p ( t ) = constant. ⇒ 1 e + ˙ ˙ c = 0 = e ( t ) + c ( t ) = constant. ⇒ 2 Matthew Douglas Johnston Guest Lecture
Basic Enzyme Model Set-up Futile Cycle (Single Equilibrium) Properties 2 -Site Phosphorylation Chain (Multiple Equilibria) Numerical Simulation What properties can we use to analyse this model? There are two conservation laws : s + ˙ ˙ c + ˙ p = 0 = s ( t ) + c ( t ) + p ( t ) = constant. ⇒ 1 e + ˙ ˙ c = 0 = e ( t ) + c ( t ) = constant. ⇒ 2 Relevant dynamics are on 2 -dimensional subspace of the original 4-dimensional space. (Variable substitution.) Matthew Douglas Johnston Guest Lecture
Basic Enzyme Model Set-up Futile Cycle (Single Equilibrium) Properties 2 -Site Phosphorylation Chain (Multiple Equilibria) Numerical Simulation What properties can we use to analyse this model? There are two conservation laws : s + ˙ ˙ c + ˙ p = 0 = s ( t ) + c ( t ) + p ( t ) = constant. ⇒ 1 e + ˙ ˙ c = 0 = e ( t ) + c ( t ) = constant. ⇒ 2 Relevant dynamics are on 2 -dimensional subspace of the original 4-dimensional space. (Variable substitution.) Quasi-steady state approximation may further reduce system to 1-dimensional space. (But with some loss of information.) Matthew Douglas Johnston Guest Lecture
Basic Enzyme Model Set-up Futile Cycle (Single Equilibrium) Properties 2 -Site Phosphorylation Chain (Multiple Equilibria) Numerical Simulation Alternative view on conservation relations... Matthew Douglas Johnston Guest Lecture
Basic Enzyme Model Set-up Futile Cycle (Single Equilibrium) Properties 2 -Site Phosphorylation Chain (Multiple Equilibria) Numerical Simulation Alternative view on conservation relations... Each reaction gives a reaction vector —a net push of each reaction in the state space of the concentrations. Matthew Douglas Johnston Guest Lecture
Basic Enzyme Model Set-up Futile Cycle (Single Equilibrium) Properties 2 -Site Phosphorylation Chain (Multiple Equilibria) Numerical Simulation Alternative view on conservation relations... Each reaction gives a reaction vector —a net push of each reaction in the state space of the concentrations. For this example, we have S + E → C C → S + E C → P + E S − 1 1 0 E − 1 1 1 1 − 1 − 1 C P 0 0 1 Matthew Douglas Johnston Guest Lecture
Basic Enzyme Model Set-up Futile Cycle (Single Equilibrium) Properties 2 -Site Phosphorylation Chain (Multiple Equilibria) Numerical Simulation Alternative view on conservation relations... Each reaction gives a reaction vector —a net push of each reaction in the state space of the concentrations. For this example, we have S + E → C C → S + E C → P + E S − 1 1 0 E − 1 1 1 1 − 1 − 1 C P 0 0 1 These vectors span a 2-dimensional subspace of the concentration space called the stoichiometric subspace (notationally, S ). Matthew Douglas Johnston Guest Lecture
Basic Enzyme Model Set-up Futile Cycle (Single Equilibrium) Properties 2 -Site Phosphorylation Chain (Multiple Equilibria) Numerical Simulation Divides state space into stoichiometric compatibility classes x 0 + S (different example pictured below): Matthew Douglas Johnston Guest Lecture
Basic Enzyme Model Set-up Futile Cycle (Single Equilibrium) Properties 2 -Site Phosphorylation Chain (Multiple Equilibria) Numerical Simulation Divides state space into stoichiometric compatibility classes x 0 + S (different example pictured below): Matthew Douglas Johnston Guest Lecture
Basic Enzyme Model Set-up Futile Cycle (Single Equilibrium) Properties 2 -Site Phosphorylation Chain (Multiple Equilibria) Numerical Simulation Divides state space into stoichiometric compatibility classes x 0 + S (different example pictured below): Roughly, more “stuff” gives a higher compatibility class (since “stuff” is usually conserved) Matthew Douglas Johnston Guest Lecture
Basic Enzyme Model Set-up Futile Cycle (Single Equilibrium) Properties 2 -Site Phosphorylation Chain (Multiple Equilibria) Numerical Simulation Without simplification, what is the long-term behavior of the system? Matthew Douglas Johnston Guest Lecture
Basic Enzyme Model Set-up Futile Cycle (Single Equilibrium) Properties 2 -Site Phosphorylation Chain (Multiple Equilibria) Numerical Simulation Without simplification, what is the long-term behavior of the system? Network structure (and intuition) dictates that S is converted into P (in some limiting way). Matthew Douglas Johnston Guest Lecture
Basic Enzyme Model Set-up Futile Cycle (Single Equilibrium) Properties 2 -Site Phosphorylation Chain (Multiple Equilibria) Numerical Simulation Without simplification, what is the long-term behavior of the system? Network structure (and intuition) dictates that S is converted into P (in some limiting way). Mathematically , we have that s + ˙ ˙ c = − k 2 c < 0 p = k 2 c > 0 . ˙ That is, we lose S and C to P as time passes. Matthew Douglas Johnston Guest Lecture
Basic Enzyme Model Set-up Futile Cycle (Single Equilibrium) Properties 2 -Site Phosphorylation Chain (Multiple Equilibria) Numerical Simulation Concentrations vs Time 2 1.8 1.6 1.4 1.2 1 0.8 [S]ubstrate 0.6 [E]nzyme [C]ompound 0.4 [P]roduct 0.2 0 0 1 2 3 4 5 6 7 8 9 10 time Figure: Numerical simulation of simple Enzyme model Matthew Douglas Johnston Guest Lecture
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