Math 609: Mathematical Methods for Systems Biology Guest Lecture - - PowerPoint PPT Presentation

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 Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties 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 Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

Basic Michaelis-Menten Enzyme Model is S + E

k+

1

⇄

k−

1

C

k2

→ P + E 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 k2 are (positive) rate constants

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

Dynamics (mass-action model) given by: ˙ s = −k+

1 s · e + k− 1 c

˙ e = −k+

1 s · e + (k− 1 + k2)c

˙ c = k+

1 s · e − (k− 1 + k2)c

˙ p = k2c where s = [S], e = [E], c = [C], and p = [P].

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

Dynamics (mass-action model) given by: ˙ s = −k+

1 s · e + k− 1 c

˙ e = −k+

1 s · e + (k− 1 + k2)c

˙ c = k+

1 s · e − (k− 1 + k2)c

˙ p = k2c 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 Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

What properties can we use to analyse this model?

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

What properties can we use to analyse this model? There are two conservation laws:

1

˙ s + ˙ c + ˙ p = 0 = ⇒ s(t) + c(t) + p(t) = constant.

2

˙ e + ˙ c = 0 = ⇒ e(t) + c(t) = constant.

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

What properties can we use to analyse this model? There are two conservation laws:

1

˙ s + ˙ c + ˙ p = 0 = ⇒ s(t) + c(t) + p(t) = constant.

2

˙ e + ˙ c = 0 = ⇒ e(t) + c(t) = constant. Relevant dynamics are on 2-dimensional subspace of the original 4-dimensional space. (Variable substitution.)

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

What properties can we use to analyse this model? There are two conservation laws:

1

˙ s + ˙ c + ˙ p = 0 = ⇒ s(t) + c(t) + p(t) = constant.

2

˙ e + ˙ c = 0 = ⇒ e(t) + c(t) = constant. 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 Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

Alternative view on conservation relations...

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties 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 Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties 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 E C P     −1 −1 1         1 1 −1         1 −1 1    

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties 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 E C P     −1 −1 1         1 1 −1         1 −1 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 Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

Divides state space into stoichiometric compatibility classes x0 + S (different example pictured below):

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

Divides state space into stoichiometric compatibility classes x0 + S (different example pictured below):

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

Divides state space into stoichiometric compatibility classes x0 + 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 Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

Without simplification, what is the long-term behavior of the system?

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties 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 Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties 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 = −k2c < 0 ˙ p = k2c > 0. That is, we lose S and C to P as time passes.

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

1 2 3 4 5 6 7 8 9 10 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Concentrations vs Time time [S]ubstrate [E]nzyme [C]ompound [P]roduct

Figure: Numerical simulation of simple Enzyme model

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties 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 Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

Consider now the Goldbeter-Koshland model (also called the futile cycle): S + E

k+

1

⇄

k−

1

C1

k2

→ P + E P + F

k+

3

⇄

k−

3

C2

k4

→ S + F

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

Consider now the Goldbeter-Koshland model (also called the futile cycle): S + E

k+

1

⇄

k−

1

C1

k2

→ P + E P + F

k+

3

⇄

k−

3

C2

k4

→ S + F Notice that different enzymes catalyze the forward and backward directions!

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

Dynamics (mass-action model) given by: ˙ s = −k+

1 s · e + k− 1 c1 + k4c2

˙ e = −k+

1 s · e + (k− 1 + k2)c1

˙ c1 = k+

1 s · e − (k− 1 + k2)c1

˙ p = k2c1 − k+

3 p · f + k− 3 c2

˙ f = −k+

3 p · f + (k− 3 + k4)c2

˙ c2 = k+

3 p · f − (k− 3 + k4)c2

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

Dynamics (mass-action model) given by: ˙ s = −k+

1 s · e + k− 1 c1 + k4c2

˙ e = −k+

1 s · e + (k− 1 + k2)c1

˙ c1 = k+

1 s · e − (k− 1 + k2)c1

˙ p = k2c1 − k+

3 p · f + k− 3 c2

˙ f = −k+

3 p · f + (k− 3 + k4)c2

˙ c2 = k+

3 p · f − (k− 3 + k4)c2

6-dimensional system with 6 undetermined parameters. Ack!

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

How can we simplify this model?

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

How can we simplify this model? Three conservation laws:

1

˙ s + ˙ c1 + ˙ c2 + ˙ p = 0 . = ⇒ s(t) + c1(t) + c2(t) + p(t) = constant.

2

˙ e + ˙ c1 = 0 = ⇒ e(t) + c1(t) = constant.

3

˙ f + ˙ c2 = 0 = ⇒ f (t) + c2(t) = constant.

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

How can we simplify this model? Three conservation laws:

1

˙ s + ˙ c1 + ˙ c2 + ˙ p = 0 . = ⇒ s(t) + c1(t) + c2(t) + p(t) = constant.

2

˙ e + ˙ c1 = 0 = ⇒ e(t) + c1(t) = constant.

3

˙ f + ˙ c2 = 0 = ⇒ f (t) + c2(t) = constant. Reduces system to 3-dimensional system.

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

How can we simplify this model? Three conservation laws:

1

˙ s + ˙ c1 + ˙ c2 + ˙ p = 0 . = ⇒ s(t) + c1(t) + c2(t) + p(t) = constant.

2

˙ e + ˙ c1 = 0 = ⇒ e(t) + c1(t) = constant.

3

˙ f + ˙ c2 = 0 = ⇒ f (t) + c2(t) = constant. Reduces system to 3-dimensional system. Quasi-steady state approximation reduces further to 1-dimension. (Loss of information.)

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

Without simplification, what kind of dynamical properties does this model have?

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

Without simplification, what kind of dynamical properties does this model have? Consider network structure:

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

Without simplification, what kind of dynamical properties does this model have? Consider network structure: . First component: S → P . Second component: P → S . = ⇒ Dynamic balance should be struck!

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

Without simplification, what kind of dynamical properties does this model have? Consider network structure: . First component: S → P . Second component: P → S . = ⇒ Dynamic balance should be struck! Questions:

1 Is this point of balance unique? 2 Is this point attracting?

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

Without simplification, what kind of dynamical properties does this model have? Consider network structure: . First component: S → P . Second component: P → S . = ⇒ Dynamic balance should be struck! Questions:

1 Is this point of balance unique? (Yes! [1], 2008) 2 Is this point attracting? (Yes! [1], 2008)

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

1 2 3 4 5 6 7 8 9 10 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Concentrations vs Time time [S]ubstrate [E]nzyme 1 [C]ompound 1 [P]roduct [E]nzyme 2 [C]ompound 2 1 2 3 4 5 6 7 8 9 10 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Concentrations vs Time time [S]ubstrate [E]nzyme 1 [C]ompound 1 [P]roduct [E]nzyme 2 [C]ompound 2

Figure: Two simulations of futile cycle with different intial conditions (same parameter values). Notice different transient behavior but same eventual long-term behavior.

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties 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 Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

Generalize the model again! (2-site phosphorylation chain): S0 + E

k+

1

⇄

k−

1

C1

k2

→ S1 + E

k+

3

⇄

k−

3

C2

k4

→ S2 + E S2 + F

k+

5

⇄

k−

5

C3

k6

→ S1 + F

k+

7

⇄

k−

7

C4

k8

→ S0 + F

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

Generalize the model again! (2-site phosphorylation chain): S0 + E

k+

1

⇄

k−

1

C1

k2

→ S1 + E

k+

3

⇄

k−

3

C2

k4

→ S2 + E S2 + F

k+

5

⇄

k−

5

C3

k6

→ S1 + F

k+

7

⇄

k−

7

C4

k8

→ S0 + F Imagine S0, S1, S2 are phosphorylated substrates, E is a kinase (adds phosphate group), F is a phosphotase (removes phosphate group).

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

Generalize the model again! (2-site phosphorylation chain): S0 + E

k+

1

⇄

k−

1

C1

k2

→ S1 + E

k+

3

⇄

k−

3

C2

k4

→ S2 + E S2 + F

k+

5

⇄

k−

5

C3

k6

→ S1 + F

k+

7

⇄

k−

7

C4

k8

→ S0 + F Imagine S0, S1, S2 are phosphorylated substrates, E is a kinase (adds phosphate group), F is a phosphotase (removes phosphate group). 9 species, 12 parameters, 3 conservation laws = ⇒ large system even after simplification!

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

We need to build intuitive approach to guide mathematical analysis.

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

We need to build intuitive approach to guide mathematical analysis. Consider network structure:

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

We need to build intuitive approach to guide mathematical analysis. Consider network structure: . First component: S0 → S1 → S2 . Second component: S2 → S1 → S0 . = ⇒ Dynamic balance struck?

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

We need to build intuitive approach to guide mathematical analysis. Consider network structure: . First component: S0 → S1 → S2 . Second component: S2 → S1 → S0 . = ⇒ Dynamic balance struck? Questions:

1 Is the point of balancing unique? 2 Is the point of balancing attracting?

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

We need to build intuitive approach to guide mathematical analysis. Consider network structure: . First component: S0 → S1 → S2 . Second component: S2 → S1 → S0 . = ⇒ Dynamic balance struck? Questions:

1 Is the point of balancing unique? 2 Is the point of balancing attracting?

More complicated than it appears...

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

For most parameter values, system settles to a dynamic equilibrium regardless of initial conditions.

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

For most parameter values, system settles to a dynamic equilibrium regardless of initial conditions.

5 10 15 20 25 30 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Concentrations vs Time (High S0) time S0 S1 S2 [E]nzyme 1 [E]nzyme 2 5 10 15 20 25 30 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Concentrations vs Time (High S1) time S0 S1 S2 [E]nzyme 1 [E]nzyme 2 5 10 15 20 25 30 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Concentrations vs Time (High S2) time S0 S1 S2 [E]nzyme 1 [E]nzyme 2

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

This is not the end of the story!

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

This is not the end of the story! In general, the long-term behavior depends on:

1 The parameter values ki, i = 1, . . . , r; 2 The initial condition x0; and 3 The stoichiometric compatibility class (spaces x0 + S).

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

This is not the end of the story! In general, the long-term behavior depends on:

1 The parameter values ki, i = 1, . . . , r; 2 The initial condition x0; and 3 The stoichiometric compatibility class (spaces x0 + S).

There are a lot of cases!

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

This is not the end of the story! In general, the long-term behavior depends on:

1 The parameter values ki, i = 1, . . . , r; 2 The initial condition x0; and 3 The stoichiometric compatibility class (spaces x0 + S).

There are a lot of cases! In fact, there are parameter values and stoichiometric compatibility classes such that there are three equilibria, two of which are stable!

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

0.5 1 1.5 2 0.1 0.2 0.3 0.4 0.5 Concentrations vs Time (High S0) time S0 S1 S2 [E]nzyme 1 [E]nzyme 2 0.5 1 1.5 2 0.1 0.2 0.3 0.4 0.5 Concentrations vs Time (High S1) time S0 S1 S2 [E]nzyme 1 [E]nzyme 2 0.5 1 1.5 2 2.5 3 0.1 0.2 0.3 0.4 0.5 0.6 Concentrations vs Time (Mid S2) time S0 S1 S2 [E]nzyme 1 [E]nzyme 2 0.5 1 1.5 2 2.5 3 0.1 0.2 0.3 0.4 0.5 0.6 Concentrations vs Time (High S2) time S0 S1 S2 [E]nzyme 1 [E]nzyme 2

Figure: Two simulations of 2-site phosphorylation chain with two different

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

Bistability in 2-site Phosphorylation Chain first observed by Markevich et al. [2] (2004).

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

Bistability in 2-site Phosphorylation Chain first observed by Markevich et al. [2] (2004). Explicit characterization of the parameter region required for bistability given in Holstein et al. [3] (2013).

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

Bistability in 2-site Phosphorylation Chain first observed by Markevich et al. [2] (2004). Explicit characterization of the parameter region required for bistability given in Holstein et al. [3] (2013). General n-site phosphorylation chain known to exhibit multistationarity for all n ≥ 2 (maximum bounded between n and 2n − 1 steady states) (Wang et al. [4] (2008).

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

Bistability in 2-site Phosphorylation Chain first observed by Markevich et al. [2] (2004). Explicit characterization of the parameter region required for bistability given in Holstein et al. [3] (2013). General n-site phosphorylation chain known to exhibit multistationarity for all n ≥ 2 (maximum bounded between n and 2n − 1 steady states) (Wang et al. [4] (2008). Multistationarity (i.e. existence of two asymptotically stable fixed points) still an active area of research.

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

These results are a part of a largest field of study know as Chemical Reaction Network Theory (CRNT).

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

These results are a part of a largest field of study know as Chemical Reaction Network Theory (CRNT). Focuses on the connection between network structure and dynamics.

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

These results are a part of a largest field of study know as Chemical Reaction Network Theory (CRNT). Focuses on the connection between network structure and dynamics. Combines dynamical systems theory, graph theory, linear algebra, algebraic geometry, biochemistry, etc. etc. etc.

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

These results are a part of a largest field of study know as Chemical Reaction Network Theory (CRNT). Focuses on the connection between network structure and dynamics. Combines dynamical systems theory, graph theory, linear algebra, algebraic geometry, biochemistry, etc. etc. etc. General, no model reduction / simplification. (Scary!)

Matthew Douglas Johnston Guest Lecture

Basic Enzyme Model Futile Cycle (Single Equilibrium) 2-Site Phosphorylation Chain (Multiple Equilibria) Set-up Properties Numerical Simulation

Selected Bibliography

David Angeli and Eduardo Sontag. Translation-invariant monotone systems, and a global convergence result for enzymatic futile cycles. Nonlinear Analysis Series B: Real World Applications, 9:128–140, 2008. Nick I. Markevich and Jan B. Hoek and Boris N. Kholodenko. Signaling switches and bistability arising from multisite phosphorylation in protein kinase cascades.

• J. Cell. Biol., 164(3):353–359, 2004.

Katharina Holstein and Dietrich Flockerzi and Carsten Conradi. Multistationarity in Sequentially Distributed Multisite Phosphorylation Networks. Available on the ArXiv at arxiv:1304.6661, 2013. Liming Wang and Eduardo Sontag. On the number of steady states in a multiple futile cycle.

• J. Math. Biol., 57:25–52, 2008.

Matthew Douglas Johnston Guest Lecture