BMED 3520 Analysis of Dynamic Models Book Chapter

BMED 3520 Analysis of Dynamic Models Book Chapter 4

Recap – Model Development • Define the problem: - processes - variables • Construct a network diagram • Convert the scheme into math model • Parameterize the model (available data) • Run simulations & analyze results (predicted reality, understand behavior) - birth, infection, recover, … S, I, R

Overview Assume that model has been constructed and parameterized What exactly should we analyze? Coarse answer: Four aspects (two modeling phases, two classes of features) Diagnostics* Analysis / Model Use Static Sensitivities Steady-state Approach Dynamic Trajectories What-If? Possible, likely, best, worst scenarios * Discovery of problems may trigger model refinement Note 1: Diagnostics and analysis not always clearly discernible Note 2: Static and dynamic features may blend into each other

Typical Dynamic Analyses Analysis of Trajectories (time courses starting from initial point) Analysis seldom possible purely with means of algebra and calculus Why? Explicit solutions (function of time vs. ODE) cannot be obtained How does the computer do it then? Numerical inching forward (e. g. , Euler Method) Linear ODE systems: Laplace transforms (engineering; seldom in biology)

Typical Dynamic Analyses Practical considerations: Usually a good idea to start simulation at the “normal” steady state Very beneficial to have a baseline scenario for comparisons e. g. , normal physiology or: typical pathology, used for testing consequences of treatments

Typical Dynamic Analyses Typical Targeted (Simulation) Experiments Change in initial value Change in independent variable Bolus experiment Change in parameter value In each case, ask: What situation does that model correspond to? What is the observed outcome? Exploration of range of possible model behaviors Monte-Carlo simulation

Interesting Dynamic Behaviors Bistability: Chapter 9 Hysteresis: Chapter 9 Phase Plane Analysis: Chapter 10 Structural Stability: System response changes qualitatively (Recall intro class; also in later chapters) 1. 5 3 X 1 X 2 X 1 0. 75 1. 5 X 1 X 2 0 0 30 time 60 0 0 120 time 240

Comments on Complex System Behaviors It does not take much to create complex behaviors (2 or 3 equations) >> “zoo of dynamical behaviors” Implication: Large systems can be much more complicated >> True to some degree But: large biological systems often buffer against behaviors that are unpredictable and/or hard to control Reality: oscillations or chaos within strongly controlled boundaries >> breathing patterns

Example of an Exceptional Nonlinear Case Lotka-Volterra Models of Interacting Populations (last class)

Recall Overview Assume that model has been constructed and parameterized What exactly should we analyze? Coarse answer: Four aspects (two modeling phases, two classes of features) Diagnostics* Analysis / Model Use Static Sensitivities Steady-state Approach Dynamic Trajectories What-If? Possible, likely, best, worst scenarios * Discovery of problems may trigger model refinement Note 1: Diagnostics and analysis not always clearly discernible Note 2: Static and dynamic features may blend into each other

Static Features Steady State (crucially important) V i+ Xi [More or less synonym: fix point, fixed point, equilibrium] Definition: (X is vector of all dep. vars. ) Steady state important for mathematical and biological reasons Many systems operate at a stable steady state homeostasis allostasis V i–

Steady State Does a steady state always exist? No Simple counterexample: 20 Function of x exp(x) 10 0. 5 x 0 1. 5 3

Steady States If a steady state exists, is it always unique? PLAS simulation: No

Steady State Steady states may not be computable algebraically, although they exist Simple, similar example as before: 20 Function of x exp(x) What can be done? 10 4 x Numerical root finding Integration of ODE 0 Issues of the above two? Difficult to catch all 0 Steady states of most realistic biological systems cannot be computed algebraically x 1. 5 3

Algebraically Characterizable Steady States Most prominent example: Linear system u is an input Computation of steady states with linear algebra (matrix algebra) Note: Although the ODEs are linear, the solutions are typically not. Example:

Example of an Exceptional Nonlinear Case Lotka-Volterra System Computation of steady states; for each equation (i): Trivial steady state: all (or some) X’s = 0 Nontrivial steady state: divide by Xi; result: linear equations Steady state characterized by linear equations, although system nonlinear

Second Rare Example of a Nonlinear Case Biochemical Systems Theory (last class): S-system Form: Vi 1+ Vi 1– Xi Vi, p+ Vi, q– Generalized Mass Action Form (GMA):

Second Rare Example of a Nonlinear Case Steady state of an S-system (does not work for GMA systems!) Computation of steady states (ignore independent variables): Define yi = ln(Xi), aij = gij – hij for all i, j and bi = ln(bi/ai) yj = ln(Xj) Steady state characterized by linear equations in logs, although system nonlinear

Models whose Steady State can be computed Definition of steady state: Linear-system Lotka-Volterra Models S-system

Stability Analysis Generic questions to be assessed: Will the system return to St St from a small perturbation in an X? Will the system return to St St from a large perturbation in an X? Typical methods: Eigenvalue analysis, numerical simulations Terminology: “System is (un)stable at a St St” or “St St is (un)stable” Not: “System is (un)stable”

Stability Analysis Answer to key question: Will the system return to St St from a small perturbation in an X?

Analysis of Steady-State Features Fundamental theorem for nonlinear systems (Hartman and Grobman): For “nice” systems, the linearized system has essentially the same features as the nonlinear system in the vicinity of the operating point. Consequence: Allows us to analyze linearized system (in the tangent plane) Tangent Plane C 1 T 3 T 1 P C 2 T 2 Analogous result for high dimensions C 3

Linearization of Nonlinear Systems For a nonlinear system with two variables, let’s assume we know one of its steady state, and we want to study the stability of that steady state. Assume the steady state is the operating point OP (XOP, YOP) Perturb the state of the system in the vicinity of OP: (XOP + ∆X, YOP + ∆Y)

Linearization of Nonlinear Systems Use Taylor approximation constants OP is steady state

Linearization of Nonlinear Systems SIR model, steady state, S=140, I=150, R=750. Study the stability of this steady state. Dynamics of original nonlinear system around OP Assume the steady state is the operating point OP. Linearize the system: Dynamics of linearized system around 0 Dynamics of linearized system shifted by OP of original system Dynamics of a nonlinear system around an OP is very similar to the dynamics of its linearized system close to OP

Stability Analysis Generic answers to previous questions: Will the system return to St St from a small perturbation in an X? >> eigenvalue analysis Will the system return to St St from a large perturbation in an X? >> difficult! Simulations are the best bet Will the system return to St St from a small perturbation in a p? >> no it won’t; but it may approach another St St Will the system return to St St from a large perturbation in a p? >> difficult! Simulations are the best bet

Stability Analysis Mathematical assessment: Compute eigenvalues (real numbers or complex conjugate pairs) Rules and observations: If all eigenvalues have negative real parts, the system is stable at the analyzed St St. Even if only one out of n real parts is positive, the system is unstable at the analyzed St St. Real parts equal zero: More analysis necessary; St St often marginally stable Pairs of non-zero imaginary parts are a sign of oscillations

Stability of a Linear System with initial condition: Solution:

Stability Analysis of 2 d System Example of 2 -dim system with no input: Linearize at st st as OP Trace: tr(A) = A 11+A 22 Determinant: det(A) = A 11 A 22 – A 12 A 21 Discriminant: d(A) = tr(A)2 – 4 det(A)

Stability Analysis Distinct (finite number) of possibilities:

Stability Analysis Example: stable spiral (A 11, A 12, A 21, A 22) = = (-3, -6, 2, 1) X 1(0) = X 2(0) = 6 X 2 10 tr(A) = -2 det(A)=9 0 “phase plane” -10 0 X 1 10

Stability Analysis Example: stable spiral (cont’d) Start system from different initial values Superimpose plots in phase plane:

(Parameter) Sensitivity Analysis Like stability, based on the linearized system Truly valid only for infinitesimally small changes Actual small changes (few percent) usually o. k. , but no guarantee Three definitions: absolute sensitivity relative sensitivity log gain If derivatives are difficult to compute, introduce numerical change of +1%

Sensitivity Analysis Example: X 1' = 4 X 2^-0. 6 - 2 X 1^. 5 X 2' = 2 X 1^. 5 - X 2^. 7 X 1 = 6 X 2 = 3 t 0 = 0 tf = 10 hr =. 01 Pathway Structure? X 1 X 2 (Not unique; e. g. , could be a cascade)

Sensitivity Analysis PLAS Analysis Steady state: (X 1, X 2) = (1. 112531, 2. 904846) Parameter sensitivities: X 1 X 2 alpha(1) 1. 07692 0. 76923 beta(1) -1. 07692 -0. 76923 alpha(2) -0. 92308 0. 76923 beta(2) 0. 92308 -0. 76923 g(1, 2) -0. 68905 -0. 49218 h(1, 1) -0. 05742 -0. 04101 Meaning?

Sensitivity Analysis Validation Experiment PLAS: Change beta(2) from 1 to 1. 1 (+10%) New steady state: (X 1, X 2) = (1. 214845, 2. 699495) DX / X 2 orig = (X 2 new - X 2 orig) / X 2 orig = - 0. 07069256 (relative change) Observed Drop of ≈ 7. 1 % Therefore, 10% increase in beta(2) leads to 7. 1% of drop in X 2. The ratio is -7. 1% / 10% = -0. 71, close to sensitivity results from PLAS X 1 X 2 alpha(1) 1. 07692 0. 76923 beta(1) -1. 07692 -0. 76923 alpha(2) -0. 92308 0. 76923 beta(2) 0. 92308 -0. 76923 g(1, 2) -0. 68905 -0. 49218 h(1, 1) -0. 05742 -0. 04101

Sensitivity Analysis Validation Experiment PLAS: Change beta(2) from 1 to 1. 01 (+1%) New steady state: (X 1, X 2) = (1. 122797, 2. 882697) DX / X 2 orig = (X 2 new - X 2 orig) / X 2 orig = - 0. 0076248 (relative change) Observed Drop of ≈ 0. 76248 % Therefore, 1% increase in beta(2) leads to 0. 76248% of drop in X 2. The ratio is -0. 76248% / 1% = -0. 76248, closer to sensitivity results from PLAS X 1 X 2 alpha(1) 1. 07692 0. 76923 beta(1) -1. 07692 -0. 76923 alpha(2) -0. 92308 0. 76923 beta(2) 0. 92308 -0. 76923 g(1, 2) -0. 68905 -0. 49218 h(1, 1) -0. 05742 -0. 04101

Sensitivity Analysis Validation Experiment PLAS: Change beta(2) from 1 to 1. 001 (+0. 1%) New steady state: (X 1, X 2) = (1. 113558, 2. 902613) DX / X 2 orig = (X 2 new - X 2 orig) / X 2 orig = - 0. 00076872 (relative change) Observed Drop of ≈ 0. 076872% Therefore, 0. 1% increase in beta(2) leads to 0. 076872% of drop in X 2. Ratio is - 0. 076872% / 0. 1% = -0. 76872, closer to sensitivity results from PLAS Infinitesimal quantity, but actual small changes (few percent) usually o. k. X 1 X 2 alpha(1) 1. 07692 0. 76923 beta(1) -1. 07692 -0. 76923 alpha(2) -0. 92308 0. 76923 beta(2) 0. 92308 -0. 76923 g(1, 2) -0. 68905 -0. 49218 h(1, 1) -0. 05742 -0. 04101

Sensitivity Analysis Log Gain (effect of change in an independent variable) X 1' = 2 X 3 X 2^-0. 6 - 2 X 1^. 5 X 2' = 2 X 1^. 5 - X 2^. 7 X 1 = 6 X 2 = 3 X 3 = 2 && X 3 t 0 = 0 tf = 10 hr =. 01 Steady state: (X 1, X 2) = (1. 112531, 2. 904846) (same as before)

Sensitivity Analysis Sensitivity analysis in PLAS: X 3 X 1 1. 07692 X 2 0. 76923 Change X 3 from 2 to 1. 90 (corresponds to 5%) Use PLAS to compute new steady state value of X 1 original(exp) X 1 new(exp) % change = 1. 112531 = 1. 052743 = (X 1 new(exp) - X 1 original(exp) ) / X 1 original(exp) = -0. 0537 = -5. 37% Sensitivity analysis: -5% change in X 3 lead to -5. 37% of change in X 1 Ratio of the two percentages is -5. 37% / -5% = 1. 0748, close to what PLAS generates

Sensitivity and Gain Analysis Note: Quality of results depends on how much the nonlinear system diverges from linearization at the OP (steady state). Considerable practical problems if steady state is difficult to compute. Easy for Linear, LV, and S-systems, hard for most other models

Other Type of Analysis: Optimization Achieve “best possible scenario” for a (nonlinear) system Examples: Maximal yield (metabolic engineering) Maximal production flux (metabolic engineering) Minimal error between model and data (parameter estimation) In most cases of metabolic engineering, operation at St St (but not always) Almost always many constraints Particularly feasible model frameworks: Stoichiometric systems S-systems (because steady state can be computed easily)

Summary Analysis of two core features: dynamic responses and steady-state properties Dynamic analysis typically through simulations Some features (e. g. , structural stability) borderline between dynamic and static Steady state very important for mathematical and biological reasons Many systems operate at a steady state (homeostasis; allostasis) Computation not always trivial; exceptions: linear, LV, S-systems Characterization easy if St St can be computed explicitly Stability, sensitivities, (log) gains represent properties of a steady state Constrained optimization possible if St St is explicit