Dynamical models

Ordinary Differential Equations

• ODE models are typically most useful when we already have an idea of the system components
  • As opposed to data-driven approaches when we don’t know how to connect the data
  • Incredibly powerful for making specific predictions about how a system works
• Limits of these approaches:
  • Results can be extremely sensitive to missing components or model errors
  • Can quickly explode in complexity
  • May rely on variables that are impossible to measure

• ODEs are extremely valuable
• During fitting, ML models are essentially a dynamic process

Applications of ODE models

Molecular kinetics

Remember BE 110!

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

\[ L_f + L_f \leftrightarrow L_D \]

\[ L_D + R_f \leftrightarrow R_b \]

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

• \(\frac{dL_f}{dt} = -k_1 L_f^2 + k_{-1}L_D\)
• \(\frac{dL_D}{dt} = k_1 L_f^2 - k_2 L_D R_f + k_{-2} R_b\)
• \(\frac{dR_b}{dt} = k_2 L_D R_f - k_{-2} R_b\)
• \(\frac{dR_f}{dt} = k_{-2} R_b - k_2 L_D R_f\)
• What could we measure here?
  • How would that be implemented?
• Function-wise what do we need here?

Pharmacokinetics

flowchart LR
    drugInjection("drug injection") --> C1
    
    subgraph CentralCompartment["central compartment"]
        C1["C<sub>1</sub><br/>V<sub>1</sub>"]
    end
    
    subgraph PeripheralCompartment["peripheral compartment"]
        C2["C<sub>2</sub><br/>V<sub>2</sub>"]
    end
    
    C1 -->|k1| C2
    C2 -->|k2| C1
    C1 -->|ke| Elimination

    style CentralCompartment rect
    style PeripheralCompartment rect

• Copy to board
• What does each compartment represent?
• \(V_1 \frac{dC_1}{dt} = k_e C_1 V_1 - k_1 C_1 V_1 + k_2 C_2 V_2\)
• \(V_2 \frac{dC_2}{dt} = k_1 C_1 V_1 + k_2 C_2 V_2\)

Applications of ODE models: Pharmacokinetics

• Central compartment corresponds to the plasma in the body.
  • \(V_1\) is the distribution volume of plasma in the body.
  • \(C_1\) is the concentration of drug in the plasma.
• Peripheral compartment represents a group of organs that significantly take up the particular drug.
  • \(V_2\) is the volume of these group of organs.
  • \(C_2\) is the concentration of drug in the group of organs.

Applications of ODE models: Pharmacokinetics

• \(k_e\) is the rate constant for clearance.
  • \(k_e C_1 V_1\) is the mass of drug/time that’s cleared.
• \(k_1\) is the rate constant for mass transfer from the central to peripheral compartment.
  • \(k_1 C_1 V_1\) is the mass of drug/time that transfers from the central to peripheral compartment.
• \(k_2\) is the rate constant for mass transfer from the peripheral to central compartment.
  • \(k_2 C_2 V_2\) is the mass of drug/time that transfers from the peripheral to the central compartment.

Applications of ODE models: Pharmacokinetics

• Have a bolus i.v. injection
  • No drug in both compartments for \(t<0\)
  • D μg of drug administered at once at \(t=0\)
  • Drug distribution occurs instantaneously in the central compartment.
    • Also get well-mixed instantaneously.
    • Concentration in central compartment at \(t=0\) is D μg/mL
• No chemical reactions in the compartment

Work out equations above. Going to learn a mathematical trick to solve!

Applications of ODE models: Population kinetics

Lotka-Volterra Equations

• maybe come back to these at the end
• Prey is x
• Predator is y
• \(\dot{x} = \alpha x - \beta x y\)
• \(\dot{y} = \delta x y - \gamma y\)

Types of Analysis

Note about difference from other models we’ve covered

• ODE models can be part of inference techniques just as elsewhere
• But we often don’t have a symbolic expression of the answer
  • Have to simulate the model every time
  • Can only focus on the input-output we get from the black box
• In this respect, what we do with ODE models will be very similar to what you could do with any computational simulation

• If we have a symbolic integral, then fitting an ODE model to data is just non-linear least squares

Analytic vs Numerical Modeling

Analytic

• Wider range of parameters
• Avoid numerical problems
• Physical intuition more direct
• Often must simplify model

Numerical

• Can handle complex models
• Dependence on parameters & initial conditions
• Physical insight may be difficult to extract
• Convergence, numerical stability

Analytic vs Numerical Modeling

Reality often requires handling in between:

• Use analytic treatment to study entire parameter space
• Use numerical treatment to study interesting regions
• Use both to handle complex behavior

Stability Analysis

• Can solve for steady-states of a system \[\frac{\delta F}{\delta t} = 0\]
• Results of this can be both stable or unstable points
  • With stable points, slope of \(\frac{\delta F}{\delta t}\) is negative
  • In multivariate case, this means eigenvalues of Jacobian are negative
• Steady-state points aren’t necessarily realistic or feasible!
  • NNLSQ can solve for points
  • Only simulating system ensures they are accessible

Walk through 1D case of stability.

Generalization

• Linear models are easier to simulate and understand than non-linear
  • Linearity: If \(\mathbf{x}_1\) and \(\mathbf{x}_2\) are both solutions, then \(c_1 \mathbf{x}_1 + c_2 \mathbf{x}_2\) is also a solution
• Linear systems tend to be separable (effective decoupling)
• Non-linear systems exhibit interesting properties

• Go through how linear systems come to a solution.
• \(\frac{dx'}{dt} = c_1 x\)
• What is this like if \(c_1\) is negative? Positive? Imaginary?

Linearity & Separability

• What is a Jacobian?
• What is steady-state here?
• Combined with above relationship, this means we can solve linear systems as a special case

Phase Portraits

\[ \dot{\vec{x}} = \vec{f}(\vec{x}) \]

Non-linear systems

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

Features in Phase Portraits

• Fixed Points (A, B, C). Steady states
• Closed Orbits (D). Periodic solutions
• Flow patterns in trajectory (A and C are similar to each other, different from B)
• Stability of fixed points and closed orbits (A, B, and C are unstable, D is stable)

Solving a Set of Equations for Phase Portrait

• Numerical computation
  • i.e., Runge-Kutta integration
• Qualitative
  • Sufficient for some purposes
• Analytic/symbolic
  • Elegant, though not always tractable

Step 1: Find the fixed points

Step 2: Determine stability of fixed points

• If the systems moves slightly away from each fixed point, will it return or will it move further away?
• Another way to ask the same question is to ask whether, as time approaches infinity, does the system tend toward or away from a given stable point.
• Note \(y\) solution must be of form:
  • \(y = y_0 e^{-t}\) (because \(\dot{y} = \frac{dy}{dt} = -y\))
  • So \(y \rightarrow 0\) for \(t \rightarrow \infty\)
• Thus, \(\dot{x} = x + e^{-y}\) becomes \(\dot{x} \rightarrow x + 1\) for long times
  • This has exponentially growing solutions
  • Toward \(\infty\) for \(x > -1\) and \(-\infty\) for \(x < -1\)

Solution grows exponentially in at least one dimension, so it is unstable.

Step 3: Sketch nullclines

Nullclines are the sets of points for which \(\dot{x} = 0\) or \(\dot{y} = 0\), so flow is either horizontal or vertical.

Step 4: Plot flow lines

• Why do solutions partition the graph?
  • Uniqueness. To cross, you must be on the solution.
• This is a non-linear version of a saddle point.
• Can we identify saddle-like behavior in linearized version of system?

Existence & Uniqueness

Non-linear \(\dot{\mathbf{x}} = f(\mathbf{x})\) and given an initial condition.

• Existence and uniqueness of solution guaranteed if \(f\) is continuously differentiable
• Corollary: Trajectories do not intersect, because if they did, then there would be two solutions for the same initial condition at the crossing point

Linearization About Fixed Points

• Think of the Jacobian as the way to convert position to velocity.

Solving Linearized Systems

• Third row is the definition of an eigenvalue.
• So if we move in \(\vec{v}\) (remember that this is delta), \(\vec{A}\) has the same effect as \(\lambda I\).
• At stable point: \(J \vec{x} = 0 = \dot{x}\)
• Definition of an eigenvalue: \(J x = \lambda x = \dot{x}\)
  • Say this is positive?
  • Say this is negative?

Example

More Examples

Classification of Fixed Points

Relevance for Nonlinear Dynamics

• So, we have said that we can find fixed points of nonlinear dynamics, linearize about each fixed point, and characterize 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, 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.

Bifurcations

• The phase portraits we have been looking at describe the trajectory of the system for a given set of initial conditions. However, 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 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

Example: Genetic Control Network

Genetic Control Network

Griffith (1971) model of genetic control:

• x = protein concentration
• y = mRNA concentration

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

• How could we compare this to data?
• Measure steady-state
• Start at points and see bifurcation
• Measure over time

Implementation

Testing

• Many properties one can test
  • Mass balance
  • Changes upon parameter adjustment
• Good to test these before and after integration

Go through example of mass balance at deriv and solution.

Python Packages

SciPy provides two interfaces for ODE solving:

• scipy.integrate.ode
• scipy.integrate.odeint

Notes:

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

Code Example: Pendulum Oscillation

The second order differential equation for the angle \(\theta\) of a pendulum acted on by gravity with friction can be written:

\[\theta''(t) + b*\theta'(t) + c*\sin(\theta(t)) = 0\]

where b and c are positive constants, and a prime (’) denotes a derivative. To solve this equation with odeint, we must first convert it to a system of first order equations. By defining the angular velocity \(\omega(t) = \theta'(t)\), we obtain the system:

\[\theta'(t) = \omega(t)\]

\[\omega'(t) = -b*\omega(t) - c*\sin(\theta(t))\]

Pendulum Oscillation

Let y be the vector \([\theta, \omega]\). We implement this system in python as:

def pend(y, t, b, c):
    theta, omega = y
    dydt = [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

Pendulum Oscillation

For initial conditions, we assume the pendulum is nearly vertical with \(\theta(0) = \pi - 0.1\), and it initially at rest, so \(\omega(0) = 0\). Then the vector of initial conditions is

y0 = [np.pi - 0.1, 0.0]

We generate a solution 101 evenly spaced samples in the interval \(0 \leq t \leq 10\). So our array of times is:

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

Pendulum Oscillation

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

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

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

Pendulum Oscillation

import matplotlib.pyplot as plt
plt.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()

Pendulum Oscillation

Analysis Example: The Cell Cycle

You build a very simple model of cells transitioning between both phases of the cell cycle?

graph LR
    G1[G1 Phase] -->|α| G2S[G2/S Phase]
    G2S -->|β| G1
    G1 -->|γ| Death[Death]
    G2S -->|γ| Death
    
    style G1 fill:#f9d5e5,stroke:#333,stroke-width:2px
    style G2S fill:#a1e7e6,stroke:#333,stroke-width:2px
    style Death fill:#f8b195,stroke:#333,stroke-width:2px

Can this model oscillate under physically meaningful situations and, if so, when?

\[\frac{dG1}{dt} = - \alpha [G1] + 2 \beta [G2] - \gamma [G1]\]

\[\frac{dG2}{dt} = \alpha [G1] - 2 \beta [G2] - \gamma [G2]\]

\[J = \begin{pmatrix} -\alpha - \gamma & 2\beta \\ \alpha & -\beta - \gamma \\ \end{pmatrix} \]

\[\lambda = \frac{1}{2} \left( -\alpha -\beta -2 \gamma \pm \sqrt{\alpha^2} + 6 \alpha \beta + \beta ^2 \right)\]

\(\alpha\) and/or \(\beta\) would have to be negative to oscillate.

Implementation Note: Stiff Systems

• Very roughly, most ODE solvers take steps inversely proportional to the rate at which the state is changing
• For systems where there are two processes operating on differing timescales, this can be problematic
  • If everything happens really fast, the system will come to equilibrium quickly
  • If everything is slow, you can take longer steps
• Stiff solvers additionally require the Jacobian matrix
  • This very roughly allows them to keep track of these differences in timescales
• odeint can automatically find this for you
  • Sometimes it’s faster/better to provide this as parameter Dfun

Implementation Note: Matrix Exponential

If \(J\) is the Jacobian matrix of an ODE model, \(y(t) = e^{Jt} y_0\).

Matrix exponential is also implemented.

• scipy.linalg.expm
  • This method is numerically stable, but there are faster implementations elsewhere.
• A commonly used package is expokit

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

Reading & Resources

• 💾: scipy.linalg.expm
• 💾: scipy.integrate.odeint
• 📖: Steven Strogatz, Nonlinear Dynamics and Chaos

Review Questions

1. What is the steady-state solution to an ODE model? How do you solve for them?
2. What is a phase-space diagram?
3. What property of ODE models ensures that solutions can never cross in phase space?
4. What is a Jacobian matrix? How do you calculate it from an ODE model?
5. What are the eigenvalues and eigenvectors of a matrix?
6. What is the behavior of an ODE system if its eigenvalues are all positive and real? How about negative and real?
7. What is the behavior of an ODE system if all its eigenvalues are imaginary and positive? How about imaginary and negative?
8. What does it mean if the eigenvalues of an ODE system give conflicting answers (e.g. one is real and negative, the other is imaginary)?
9. How can you fit an ODE model to a series of measurements over time?
10. How could you constrain an ODE system so that you only fit parameters that have oscillation?