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Week 5, Lecture 9 - Dynamical models

Aaron Meyer

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Outline

I Administrative IssuesI Review ODE models

I Some common constructionsI Classic analysis:

I Stability analysisI Pseudo-steady-state

I Implementation:I Numerical integrationI Stiff systemsI Matrix exponentials

Slides partly adapted from those by Bruce Tidor.

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Ordinary Differential Equations

I ODE models are typically most useful when we already havean idea of the system componentsI As opposed to data-driven approaches when we don’t know

how to connect the dataI Incredibly powerful for making specific predictions about how a

system worksI Limits of these approaches:

I Results can be extremely sensitive to missing components ormodel errors

I Can quickly explode in complexityI May rely on variables that are impossible to measure

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Applications of ODE models: Molecular kinetics

Remember BE100!

Let’s say we have to ligands that dimerize, then this dimer binds toa receptor as one unit:

Lf + Lf ↔ LD

LD +Rf ↔ Rb

If we want to know about how these species interact, we can modeltheir behavior with the rate equations that describe this process.

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Applications of ODE models: Pharmacokinetics

C1

V1

druginjection

ke

C2

V2

k1

k2

centralcompartment

peripheralcompartment

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Applications of ODE models: Pharmacokinetics

I Central compartment corresponds to the plasma in the body.I V1 is the distribution volume of plasma in the body.I C1 is the concentration of drug in the plasma.

I Peripheral compartment represents a group of organs thatsignificantly take up the particular drug.I V2 is the volume of these group of organs.I C2 is the concentration of drug in the group of organs.

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Applications of ODE models: Pharmacokinetics

I ke is the rate constant for clearance.I keC1V1 is the mass of drug/time that’s cleared.

I k1 is the rate constant for mass transfer from the central toperipheral compartment.I k1C1V1 is the mass of drug/time that transfers from the

central to peripheral compartment.I k2 is the rate constant for mass transfer from the peripheral

to central compartment.I k2C2V2 is the mass of drug/time that transfers from the

peripheral to the central compartment.

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Applications of ODE models: Pharmacokinetics

I Have a bolus i.v. injectionI No drug in both compartments for t<0I D µg of drug administered at once at t=0I Drug distribution occurs instantaneously in the central

compartment.I Also get well-mixed instantaneously.I Concentration in central compartment at t = 0 is D µg/mL

I No chemical reactions in the compartment

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Applications of ODE models: Population kinetics

Lotka-Volterra Equations

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Note about difference from other models we’ve covered

I ODE models can be part of inference techniques just aselsewhereI If we have a symbolic integral, then fitting an ODE model to

data is just non-linear least squaresI But we often don’t have a symbolic expression of the answer

I Have to simulate the model every timeI Can only focus on the input-output we get from the black box

I In this respect, what we do with ODE models will be verysimilar to what you could do with any computationalsimulation

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Analytic vs Numerical Modeling

I AnalyticI Wider range of parametersI Avoid numerical problemsI Physical intuition more directI Often must simplify model

I NumericalI Can handle complex modelsI Dependence on parameters & initial conditionsI Physical insight may be difficult to extractI Convergence, numerical stability

Reality often requires handling in between:

I Use analytic treatment to study entire parameter spaceI Use numerical treatment to study interesting regionsI Use both to handle complex behavior

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Stability Analysis

I Can solve for steady-states of a system

δF

δt= 0

I Results of this can be both stable or unstable pointsI With stable points, slope of δFδt is negativeI In multivariate case, this means eigenvalues of Jacobian are

negativeI Steady-state points aren’t necessarily realistic or feasible!

I NNLSQ can solve for pointsI Only simulating system ensures they are accessible

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Generalization

I Linear models are easier to simulate and understand thannon-linearI Linearity: If x1 and x2 are both solutions, then c1x1 + c2x2 is

also a solutionI Linear systems tend to be separable (effective decoupling)I Non-linear systems exhibit interesting properties

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Linearity & Separability

2

• Linear models are easier to simulate and understand than non-linear

Generalization

understand than non linear– Linearity:

If x1 and x2 are both solutions, then c1x1 + c2x2 is also a solution

– Linear systems tend to be separable (effective decoupling)

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Linearity & Separability

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Phase Portraits

3

Phase Portraits

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xxfx &&�&

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),( 2122 xxfx °¿

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• Non-linear systems– No general analytic approach to finding trajectory– So, goal is to understand qualitative trajectory behavior

Features in Phase Portraits

B

1. Fixed Points (A, B, C). Steady states

A

B

CD

2. Closed Orbits (D). Periodic solutions3. Flow patterns in trajectory

A & C are similar to each other, different from B

4. Stability of fixed points & closed orbitsA, B, & C are unstable, D is stable

Non-linear systems

I No general analytic approach to finding trajectoryI So, goal is to understand qualitative trajectory behavior

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Features in Phase Portraits

3

Phase Portraits

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• Non-linear systems– No general analytic approach to finding trajectory– So, goal is to understand qualitative trajectory behavior

B

A

B

CD

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Solving a Set of Equations for Phase Portrait

I Numerical computationI i.e., Runge-Kutta integration

I QualitativeI Sufficient for some purposes

I AnalyticI Elegant, though not always tractable

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Example – fixed points

4

Solving a Set of Equations for Phase Portrait

• Numerical computation – i e Runge-Kutta integration– i.e., Runge-Kutta integration

• Qualitative– Is often sufficient for most purposes

• Analytic– Elegant, though not always tractable

Example – fixed points

yyexx y

� � �

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Nonlinear, becausecan’t be represented as

dycxybyaxx

� �

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dycxy � Step 1: Find fixed points

Fixed points (also called stationary points) are those points where the time-derivative of each coordinate is zero. 0 and 0 yx ��

0010

� ��� ��

yyexex yy

)0,1(),(at point fixed one Thus,

00

�

��

yx

yy

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Example – stability

Step 2: Determine stability of fixed pointsI If the systems moves slightly away from each fixed point, will

it return or will it move further away?I Another way to ask the same question is to ask whether, as

time approaches infinity, does the system tend toward or awayfrom a given stable point.

I Note y solution must be of form:I y = y0e

−t (because y = dydt = −y)

I So y → 0 for t→∞I Thus, x = x+ e−y becomes x→ x+ 1 for long times

I This has exponentially growing solutionsI Toward ∞ for x > −1 and −∞ for x < −1

Thus, overall solution grows exponentially in at least one di-mension, and so is unstable.

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Example – nullclines

Step 3: Sketch nullclinesNullclines are the sets of points for which x = 0 or y = 0, so flow iseither horizontal or vertical.

5

Example – stability

Step 2: Determine stability of fixed pointsIf the systems moves slightly away from Another way to ask the same question each fixed point, will it return or will it move further away?

.for 0so),(because

formofbemust solution Note

0

foo

� �

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dtdyyeyy

yt �

is to ask whether, as time approaches infinity, does the system tend toward or

away from a given stable point.

y

)1for and 1for (towardssolutions growinglly exponentia has This times.longfor 1becomes Thus,

��f��!f�

�o� �

xx

xxexx y ��

Thus, overall solution grows exponentially in at least one dimension, and so is unstable

y

yexx

xxxyyy

�� o

�!� � o�

00for verticalis Flow

1for right the tois flow so here, 1axis-at x horizontal is flow 00for 0

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y

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yx��

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space into classes of flow direction:

¯®­¿¾½²¢ 0, yx ��

1

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Example – computed

Step 4: Plot flow lines

6

Example – computed

yStep 4: Plot flow lines

x–1

Notice that the major vertical blue flow line does not coincide with the nullcline.

This is a non-linear version of a saddle point.

Can we identify saddle-like behavior in linearized

version of system?

Existence & Uniqueness

)(xfx &&�& and given initial conditionNon-linear

• Existence & uniqueness of solution guaranteed if is continuously differentiable

• Corollary: Trajectories do not intersect, because if they did, then there would be two solutions for the

f&

y ,same initial condition at the crossing point

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Existence & Uniqueness

Non-linear x = f(x) and given an initial condition.

I Existence and uniqueness of solution guaranteed if f iscontinuously differentiable

I Corollary: Trajectories do not intersect, because if they did,then there would be two solutions for the same initialcondition at the crossing point

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Linearization About Fixed Points

7

Linearization About Fixed Points

)()(0

),(point fixed with systemlinear -non a be ),(),(Let

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yxyxgyyxfx¿¾½

¯®­

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point fixed from deviations be Let *

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Change of variable

*

),,( Likewise,

expansion seriesTaylor ),,(),(,(constant) is (

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xguv

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Solving Linearized Systems

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Solving Linearized Systems

7

Linearization About Fixed Points

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Example

8

Example

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x

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(stable) 3,41

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Saddle Point(one positive real and one negative real eigenvalue)

More ExamplesyStable Node

(two real negative eigenvalues)

ySpiral(two complex eigenvalues;x

yUnstable Node(t l iti

xeigenvalues;Can be growing or shrinking)

yCenter( l i i

x

(two real positive eigenvalues)

x

(purely imaginary eigenvalues)

Also, equal eigenvalues lead to stars & degenerate nodes

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More Examples

8

Example

4111

32

6

1

)32()(24

2

1

2

1

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x

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(stable) 3,41

22 � ¸¹·¨

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Saddle Point(one positive real and one negative real eigenvalue)

More ExamplesyStable Node

(two real negative eigenvalues)

ySpiral(two complex eigenvalues;x

yUnstable Node(t l iti

xeigenvalues;Can be growing or shrinking)

yCenter( l i i

x

(two real positive eigenvalues)

x

(purely imaginary eigenvalues)

Also, equal eigenvalues lead to stars & degenerate nodes

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Classification of Fixed Points

9

Classification of Fixed Points

WUnstable nodes

042 '�W

'

Unstable spirals

Stable spiralsSadd

le po

ints

Centers

Stable nodes

Non-isolatedfixed points Stars, degenerate nodes

Relevance for Nonlinear Dynamics

• So, we have said that we can find fixed points of nonlinear dynamics, linearize about each fixed point, and y pcharacterize the dynamics about each fixed point in the non-linear model by the corresponding linear model.

• Is this always true? Do the nonlinearities ever disturb this approach?

• A theorem can be proven which states – That all the regions on the previous slide are “robust” (nodes,That all the regions on the previous slide are robust (nodes,

spirals, saddles) and correspond between linear and nonlinear models.

– But that all the lines on the previous slide are “delicate” (centers, stars, degenerate nodes, non-isolated fixed points) and can have different behaviors in linear and non-linear models.

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Relevance for Nonlinear Dynamics

I So, we have said that we can find fixed points of nonlineardynamics, linearize about each fixed point, and characterizethe dynamics about each fixed point in the non-linear modelby the corresponding linear model.

I Is this always true? Do the nonlinearities ever disturb thisapproach?

I A theorem can be proven which statesI That all the regions on the previous slide are “robust” (nodes,

spirals, saddles) and correspond between linear and nonlinearmodels.

I But that all the lines on the previous slide are “delicate”(centers, stars, degenerate nodes, non-isolated fixed points)and can have different behaviors in linear and non-linearmodels.

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Bifurcations

I The phase portraits we have been looking at describe thetrajectory of the system for a given set of initial conditions.However, for “fixed” parameters (rate constants in eqns, forinstance).

I What we might like is a series of phase portraitscorresponding to different sets of parameters.

I Many will be qualitatively similar. The interesting ones willbe where a small change of parameters creates a qualitativechange in the phase portrait (bifurcations).

I What we will find is that fixed points & closed orbits can becreated/destroyed and stabilized/destabilized.

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Saddle-Node Bifurcation

10

Bifurcations

• The phase portraits we have been looking at describe the trajectory of the system for a given set of initial conditions. j y y gHowever, for “fixed” parameters (rate constants in eqns, for instance).

• What we might like is a series of phase portraits corresponding to different sets of parameters.

• Many will be qualitatively similar. The interesting ones will be where a small change of parameters creates abe where a small change of parameters creates a qualitative change in the phase portrait (bifurcations).

• What we will find is that fixed points & closed orbits can be created/destroyed and stabilized/destabilized.

Saddle-Node Bifurcation

yyxx

� �

�� 2P

x

y

x

y

x

y

P�> 0 P�= 0 P�< 0

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Genetic Control Network

Griffith (1971) model of genetic control:

I x = protein concentrationI y = mRNA concentration

11

Genetic Control Network

Griffith (1971) model of genetic controlx = protein concentrationy = mRNA concentrationy

byxx

y

yaxx

��

��

2

2

1�

� protein degrades and is synthesized from mRNA

mRNA degrades and is stimulated by protein dimer

DNA

mRNA (y)

protein (x)dimerization

a

b

Genetic Control Network

nullclines:

� �2

2

2

2

110

0

xbx

ybyxx

y

axyyaxx

� o�

�

o��

�

�

ychanging a with fixed bonly changes slope of blueline and can change numberof stationary points

x

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Genetic Control NetworkBiochemical version of a bistable switch:

1. Only stable points are no protein and mRNA or a fixedcomposition

2. If degradation rates too great, only stable point is origin

12

Genetic Control Network

Biochemical version of a bistable switch:(1) only stable points are no protein and mRNA or a fixed composition(2) if degradation rates too great, only stable point is origin

y

x

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Implementation - Testing

I Many properties one can testI Mass balanceI Changes upon parameter adjustment

I Good to test these before and after integration

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Implementation

SciPy provides two interfaces for ODE solving:

I scipy.integrate.odeI scipy.integrate.odeint

Notes:

I Both can solve stiff and non-stiff equations.I ode has a number of different methods. Pay attention to the

“set_integrator” option.

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Implementation - ExampleThe second order differential equation for the angle θ of a pendulumacted on by gravity with friction can be written:

θ′′(t) + b ∗ θ′(t) + c ∗ sin(θ(t)) = 0

where b and c are positive constants, and a prime (‘) denotes aderivative. To solve this equation with odeint, we must first convertit to a system of first order equations. By defining the angularvelocity ω(t) = θ′(t), we obtain the system:

θ′(t) = ω(t)

ω′(t) = −b ∗ ω(t)− c ∗ sin(θ(t))

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Implementation - Example

Let y be the vector [θ, ω]. We implement this system in python as:

def pend(y, t, b, c):theta, omega = ydydt = [omega, -b*omega - c*np.sin(theta)]return dydt

We assume the constants are b = 0.25 and c = 5.0:

b, c = 0.25, 5.0

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Implementation - Example

For initial conditions, we assume the pendulum is nearly vertical withθ(0) = π−0.1, and it initially at rest, so ω(0) = 0. Then the vectorof initial conditions is

y0 = [np.pi - 0.1, 0.0]

We generate a solution 101 evenly spaced samples in the interval0 ≤ t ≤ 10. So our array of times is:

t = np.linspace(0, 10, 101)

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Implementation - Example

Call odeint to generate the solution. To pass the parameters b andc to pend, we give them to odeint using the args argument.

from scipy.integrate import odeintsol = odeint(pend, y0, t, args=(b, c))

The solution is an array with shape (101, 2). The first column is θ(t),and the second is ω(t). The following code plots both components.

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Implementation - Example

import matplotlib.pyplot as pltplt.plot(t, sol[:, 0], 'b', label='theta(t)')plt.plot(t, sol[:, 1], 'g', label='omega(t)')plt.legend(loc='best')plt.xlabel('t')plt.grid()plt.show()

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Implementation - Example

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Implementation - Stiff Systems

I Very roughly, most ODE solvers take steps inverselyproportional to the rate at which the state is changing

I For systems where there are two processes operating ondiffering timescales, this can be problematicI If everything happens really fast, the system will come to

equilibrium quicklyI If everything is slow, you can take longer steps

I Stiff solvers additionally require the Jacobian matrixI This very roughly allows them to keep track of these

differences in timescalesI odeint can automatically find this for you

I Sometimes it’s faster/better to provide this as parameter Dfun

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Implementation - Matrix Exponential

If J is the Jacobian matrix of an ODE model, y(t) = eJty0.

Matrix exponential is also implemented.

I scipy.linalg.expmI This method is numerically stable, but there are faster

implementations elsewhere.I A commonly used package is expokit

For linear systems, this can be >1000x faster.

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Further Reading

I scipy.linalg.expmI scipy.integrate.odeintI Steven Strogatz, Nonlinear Dynamics and Chaos