1. Introduction to Dynamical Systems
Eric Sobie
Pharmacology and Systems Therapeutics
Mount Sinai School of Medicine 1

2. Outline Modeling using ODEs Analyzing stability of ODE systems
Law of mass action
Converting from a diagram to a system of equations
Euler’s method to integrate systems of ODEs
Analyzing stability of ODE systems
One-dimensional examples
Phase-plane techniques for two-dimensional systems
Example: yeast glycolytic oscillations
Nullclines, stable and unstable fixed points
Bifurcations: abrupt changes in system behavior
Workshop/Homework assignment
Implementation and analysis of two-variable ODE model 2

3. Systems of Ordinary Differential Equations
Consider ligand binding to a receptor
In the steady-state, bound receptor can be calculated as:
[L] = free ligand
[R] = free receptor
[R]TOT = total receptor
[LR] = ligand-receptor complex
[LR] (nM)
RTOT = 20 nM
[L] varies from 0 to 200 µM
KD can be 10, 30, 50, 70, 90 µM
[L] (µM)
What about non-steady-state solutions? 3

4. Systems of Ordinary Differential Equations
Ligand binding to a receptor
More generally, this reaction implies the following set of equations
Where do these equations come from? 4

5. Law of mass action Forward rate (concentration/time) = Forward rate =
The rate of an elementary reaction (a reaction that proceeds through only one transition state, that is one mechanistic step) is proportional to the product of the concentrations of the participating molecules. (source: Wikipedia)
Forward rate (concentration/time) =
Forward rate =
[A], [B], [C] = generic chemical species
Here, rate constants and have different units! 5

6. Enzyme-catalyzed reactions
Frequently shorthand is used
[Substrate]
[Product]
[S] = free substrate
[E] = free enzyme
[E]TOT = total enzyme
[P] = product
[Enzyme]
Michaelis-Menten kinetics are usually assumed
The equations in this case are as follows:
In chemical reality, this actually involves additional steps
Next we will discuss the assumptions underlying this representation 6

7. This scheme implies the following equations:
Enzyme Kinetics
This scheme implies the following equations:
Units: [E],[S],[ES],[P] (M)
k1, k-2 (M-1 s-1)
k-1, k2 (s-1) 7

8. Enzyme kinetics Note: This is not the same as assuming k-2 = 0
Four simplifications are made to derive the MM equations
These require three important assumptions
1) Examine initial rate of production only
If [P] = 0, then several terms drop out:
Note: This is not the same as assuming k-2 = 0 8

9. Enzyme kinetics 2) [S] is large compared with [E]
Thus, [S] >> [ES], and we can assume d[S]/dt = 0
3) Notice that d[E]/dt = -d[ES]/dt
This is equivalent to saying:
Remember that enzyme is neither produced nor consumed.
Note: This is not an assumption, this is a mathematical and physical fact (conservation of mass). 9

10. Because of (3), we can substitute:
Enzyme kinetics
Because of (3), we can substitute:
4) Steady-state assumption, d[ES]/dt = 0 10

11. Enzyme kinetics Now we have to solve for steady-state value of [ES].
This will let us calculate V0 = d[P]/dt = k2[ES]
Factor out [ES]
Divide top and bottom by k1
Note: The above is merely algebra 11

12. Enzyme kinetics Now we have our steady-state value of [ES] Define:
Then we obtain the famous Michaelis-Menten equation: 12

13. More complicated biochemical reactions
Example: a generic three component repressive network
this can display oscillations, depending on parameter values
Proteins A, B and C can be phosphorylated (X-p) or dephosphorylated (X)
A catalyzes dephosphorylation of B
B catalyzes dephosphorylation of C
C catalyzes phosphorylation of A
Mogilner et al., Developmental Cell 11:279–287, 2006
Note: This scheme is a simplified version of an oscillatory biochemical circuit in soil bacteria (see Igoshin et al., PNAS 101:15760–15765, 2004) 13

14. More complicated biochemical reactions
Example: a generic three-component repressive network
Equations just for [A] and [AP]
Notice that:
Same as: [A] + [AP] = [A]T
Mogilner et al., Developmental Cell 11:279–287, 2006
One equation can be eliminated 14

15. More complicated biochemical reactions
The full set of equations underlying this scheme
Mogilner et al., Developmental Cell 11:279–287, 2006 15

16. More complicated biochemical reactions
Sometimes the shorthand is even more extreme
What this really means is:
[Raf]
[MEK]
[MEK-PP]
[ERK]
[ERK-PP]
Equations can be derived from this scheme, not from scheme on the left
[Raf], [MEK], [ERK] = proteins involved in mitogen-activated protein kinase signaling
[-PP] = doubly phosphorylated form
Diagram from the RIKEN BioResource Center: 16

17. More complicated biochemical reactions
Occasionally the diagrams do not permit model-building
Kamrava et al., Oncogene 24: , 2005
No mechanism from receptor to [Ca2+]
Leaves out the MAPKKK, MAPK steps
Diagrams can frequently, but not always, be converted into ODEs 17

18. Solving ordinary differential equations
Euler's method
So, we start with x(0), which is known
Leonhard Euler
( )
Now x(Δt) is known
etc. 18

19. We can write simple MATLAB code to solve this numerically
Euler's method example
Assume a=20, b=2, c=5
We can write simple MATLAB code to solve this numerically
a = 20 ;
b = 2 ;
c = 5 ;
dt = 0.05 ;
tlast = 2 ;
iterations = round(tlast/dt) ;
xall = zeros(iterations,1) ;
x = c ;
for i = 1:iterations
xall(i) = x ;
dxdt = a - b*x ;
x = x + dxdt*dt ;
end % of this time step
time = dt*(0:iterations-1)' ;
figure
plot(time,xall) 19

20. What does this code actually do?
Euler's method example
What does this code actually do?
a=20; b=2; x0=5 5 6 7 8 9
x = 5
dx/dt = 20 – 2*5 = 10
x = *0.05 = 5.5
dx/dt = 20 – 2*5.5 = 9
x = *0.05 = 5.95
etc.
0.8
0.6
0.4
0.2 1 20

21. Euler's method example
a=20; b=2; c=5
This simple differential equation has an analytical solution
For small values of Δt, the numerical solution is accurate 21

22. Notes on Euler's method
Extending this to systems of ODEs is straightforward
Solutions can become highly unstable if Δt is too large
In MATLAB implementations, one of two things commonly happen:
variables achieve values such as 4.3 x 1078
2) Strictly non-negative variables (concentrations) become negative
In the 227 years since Euler's death, applied mathematicians have devised more stable and accurate methods. These are the algorithms implemented by MATLAB's built in solvers (e.g. ode23). 22

23. Other numerical methods for ODE systems
Euler died more than 200 years ago – since then, improvements to his algorithm have been made
Runge-Kutta method
Also compute dx/dt between t0 and t1; use a weighted average
Variable time-step methods
Small Δt required
Big Δt okay
These algorithms are available in MATLAB as the built-in ODE solvers: ode23, ode45, ode15s, ode23tb, etc. 23

24. Model structure to solve an ODE system
The Euler’s method MATLAB script is structured as follows:
1) Define constants
2) Set time step, simulation time, etc.
3) Set initial conditions
4) A "for" loop to simulate evolution of time
At each time step:
write output if needed
compute dX/dt
compute X at the next time step
5) Plot and output results 24

25. Solving ODE systems with MATLAB solvers
To use MATLAB’s solvers, the structure will be modified:
1) Define constants
2) Set time step, simulation time, etc.
3) Set initial conditions
4) Use MATLAB's solvers to integrate the system
5) Plot and output results
This will require writing a function that, given a collection of state variables, computes the set of derivatives 25

26. MATLAB scripts versus functions
Schematic relationship between scripts and functions
A B C
result = samplefunction(A,B,C) ;
samplefunction.m
result
function a = samplefunction(x,y,z)
LOTS OF IMPORTANT MATLAB CODE
a = something ;
return 26

27. MATLAB scripts versus functions
What does this have to do with solving ODEs?
The MATLAB ode solvers, e.g. "ode23," are functions. To use them properly, you need to know what variables to pass to them.
To use the MATLAB ode solvers, you must create a function that computes the derivatives of your variables.
[time,statevars] = ;
function deriv = dydt_tyson(t,statevar) 27

28. Solving ODEs with MATLAB functions
To solve the simple 1-variable ODE:
ode.m
global a b;
a = 20 ;
b = 2 ;
c = 5 ;
tlast = 4 ;
x0 = c ;
[t,x] = ;
figure
plot(t,x)
dxdt.m
function deriv = dxdt(t,x)
global a b ;
deriv = a - b*x ;
return 28

29. Analyzing stability of ODE systems
Example: a generic three-component repressive network
The scheme implies a set of differential equations
Mogilner et al., Developmental Cell 11:279–287, 2006
The equations are solved using standard numerical techniques 29

30. ODE models Two solutions to this system Parameter set 1
Parameter values greatly influence system behavior
We use tools of “dynamical systems” to understand different behaviors 30

31. Stability analysis of ODE systems
A one-dimensional example
Isolated cardiac myocyte 10 8 6
Fixed points 4
dV/dt (mV/ms) 2 -2 -4
-100
-90
-80
-70
-60
V (mV)
Change to < -85  positive dV/dt
Change to ~-70  negative dV/dt
Change to >~-58  positive dV/dt
Beginning with a myocyte at rest (-85 mV), simulate instantaneous changes in voltage, calculate Iion 31

32. Stability analysis of ODE systems
Instantaneous changes in membrane potential
dV/dt = -Iion/Cm
Change V, then integrate equations 10 8 6 4
-95 mV
dV/dt (mV/ms)
-75 mV 2
-65 mV
V (mV)
-50
-55 mV -2 -4
5 ms
-100
-100
-90
-80
-70
-60
V (mV)
Deviations from -85 mV up to ~-58 mV result in a return to the resting state.
This is a stable fixed point
Small deviations from -58 mV cause action potentials or return to the resting state.
This is an unstable fixed point
We will learn to analyze these more rigorously 32

33. Stability analysis of ODE systems
A two-dimensional example: Yeast glycolytic oscillations
Tu et al., Science 310: , 2005
Many mathematical models of this process have been developed
Bier, Bakker, & Westerhoff published a very simple one (Biophys. J. 78: , 2000)
We will analyze stability using this example system 33

34. Stability analysis of ODE systems
Bier model of yeast glycolytic oscillations
[Glucose]out
[Glucose]in
[ATP]
[ADP]
Vin V1 Vp
glucose transport
phosphofructokinase
ATPases
Default parameter values: Vin = 0.36, k1 = 0.02, kp = 6
Km = 13
Km = 20
[G]
[ATP]
How can we understand the qualitatively different behavior? 34

35. Stability analysis of ODE systems
Phase-plane techniques for 2D systems
Instead of plotting [G] and [ATP] vs. time, plot [G] vs. [ATP]
Km = 13
Km = 20 16 16 12 12
[Glucose]
[Glucose] 8 8 4 4 4
[ATP] 8 12 4
[ATP] 8 12
[G] and [ATP] oscillate indefinitely in a “stable limit cycle”
[G] and [ATP] converge to a “stable fixed point” 35

36. Stability analysis of ODE systems
It is useful to plot “nullclines”
This is the set of points for which d[G]/dt = 0; d[ATP]/dt = 0
These can usually be calculated analytically 16
d[G]/dt = 0 16
d[G]/dt = 0 12 12
[Glucose]
[Glucose]
d[ATP]/dt = 0 8 8
d[ATP]/dt = 0 4 4 4 8 12 4 8 12
[ATP]
[ATP]
Where the nullclines intersect, both derivatives are zero.
This is a “fixed point” 36

37. Stability analysis of ODE systems
What if we start the oscillating system close to the fixed point?
Km = 13 16
[G]
[ATP] 12
[Glucose] 8 4 4 8 12
[ATP]
This system moves away from the fixed point, then will oscillate forever
The fixed point is “unstable.” The oscillation is a “stable-limit cycle.” 37

38. Stability analysis of ODE systems
What if we start the non-oscillating system away from the fixed point?
Km = 20 16
[G]
[ATP] 12
[Glucose] 8 4 4
[ATP] 8 12
No matter the initial conditions, this system moves towards the fixed point
This fixed point is “stable.” 38

39. Stability analysis of ODE systems
How can we understand stable and unstable fixed points mathematically?
Compute the “Jacobian” matrix:
Evaluate this at the fixed point defined by [G]*, [T]*
(This is where analytical computations can become difficult.) 39

40. Stability analysis of ODE systems
Evaluate Jacobian matrix at the fixed point defined by [G]*, [T]*
The eigenvalues of the Jacobian (at the fixed point) determine stability
Eigenvalues can be real or complex numbers
Real parts of both are positive: the fixed point is unstable
Real parts of both are negative: the fixed point is stable
Complex eigenvalues have positive real parts: a limit cycle is stable
Other possibilities are encountered less frequently 40

41. Bifurcations
A bifurcation is where the system qualitatively changes its behavior
Here we simulate the Bier model for many different values of Km
With each simulation, plot [G]min and [G]max over last 500 minutes 30
This is the bifurcation 20
[G]min and [G]max 10 10 15 Km 20 25
At Km ≈ 16, the unstable fixed point becomes stable.
Later we will see bifurcations that are easy to visualize in phase space. 41

42. Workshop/Homework Assignment
Implement the Bier model and determine stability with different parameters
You will be provided with the script “euler.m” shown on slide 11
Alter the code so that it simulates the Bier equations
euler.m
bier.m
0.8
0.6
0.4
0.2 1 5 6 7 8 9
Homework provides step-by-step instructions
Questions involve effects of changes in phosphofructokinase activity 42

43. Slides from a lecture in the course Systems Biology—Biomedical Modeling
Citation: E. A. Sobie, An introduction to dynamical systems. Sci. Signal. 4, tr6 (2011).