Representations of Biological Processes

1. Representations of Biological Processes Eberhard O. Voit Integrative Core Problem Solving with Models September 2011

2. Overarching Challenges of Modeling in Biology Map reality into a diagram Map diagram into symbolic equations Convert symbolic equations into functional symbolic equations Convert functional symbolic equations into parameterized equations (This course: started with complete model) The rest is comparatively straightforward!

3. Specific Challenge of Designing Equations Biology is complicated No guidance given as to what functions to use E.O. Voit, J.S. Almeida, S. Marino, R. Lall, G. Goel, A.R. Neves, and H. Santos: IEE Proc. Systems Biol. 2006

4. Questions: Are there generic mathematical representations for biology? Generic: applicable to more than a few special cases If so, where do we find them? If we find one, are they unique? Without knowing “true” representations, how can we even get started? Is anything in modeling valid?

5. Example Ubiquitous S-shaped curve; e.g.: Growth of Bacterial Colony 50 40 30 Size of Colony (cm2) 20 10 0 0 1 2 3 4 5 Age (days) Is there a “best” representation? Are there several? Are they equally good? From: Torres and Voit, 2002

6. f(x) 2 f1 data 1 0 f2 f3 1 0 f4 f5 1 0 0 2 2 4 x Example for Non-Uniqueness Representations of artificial S-shaped data with modest noise: f5(x) = 0.8·sin(0.82·x+4.8) +1 From: Torres and Voit, 2002

7. Look for Guidance, Role Models Using Elementary Math: Bacterial Shape >>> One cylinder and two half-spheres

8. Using Elementary Math: Bacterial Growth (Initial Phase) Process: cell division (doubling); therefore Pt+t= 2 Pt P(t+t ) +t= 2 Pt+t = 2  2 Pt = 22Pt … with some math, approximately: Issue: Mechanisms of processes seldom known

9. Modeling in Physics: Fundamental Laws E = m c2 I = dq/dt Current = differential of electrical charge Relationship between voltage, current, capacitance Problem: Biological processes composed of myriads of physical processes that are intertwined in complicated ways Examples: G-protein coupled receptor in signal transduction Growth of a mouse Carbon cycling in a forest Reality: Complicated compound functions

10. vdP(t) 4 2 0 0 30 60 time System of non-linear differential equations Modeling in Engineering: Linear functions / linear ODE systems Rationale: Can design things so that behave ~ linearly; maybe piecewise linear Problem, demonstrated by example: Heartbeat modeled as stable limit cycle sin(t) 4 2 0 0 30 60 time System of linear differential equations

11. Details on Oscillators k = 5 v(0) = 2 w(0) = 0 System of linear differential equations System of nonlinear differential equations (van der Pol oscillator) Note that solutions of ODEs are not linear! (Sine and cosine) 11

12. Modeling in Chemistry: Mass action functions X1 k3 k X3 X Y X2 Rationale: Probability theory, Statistical mechanics, Random encounters in homogeneous space. All problems solved?

13. A+B P+Q from Schultz (1994) Use “True” Mass Action Functions in General?

14. k1 k-1 P Special Case: Michaelis-Menten Rate Law k2 E + S E S E +

15. Special Case: Michaelis-Menten Rate Law (Cont’d) Quasi-steady-state assumption: (ES) = constant >> do simple (somewhat messy) algebra; result: Vmax = k2Etotal = k2 (E + (ES))

16. Special Case: Michaelis-Menten Rate Law (Cont’d) Concentration Reaction speed Vmax 10 2 vP P S 5 1 ES 0 0 KM 5 10 0 5 10 time Concentration of S

17. Similar Rationale and Result: Hill Rate Law Reaction speed Vmax 10 5 0 10 20 Concentration of S For n = 1: MMRL >>> Uncounted variations (inhibition, activation, …)

18. Issues Mechanism valid? QSSA valid? Homogeneous space? Free movement? True for small numbers of molecules? Processes outside biochemistry?

19. Generic Alternative: Approximation Canonical Models Replace a (complicated) function with a (simpler) function 10 10 f(x) f(x) 5 5 A(x) L(x) x x -2 0 2 -2 0 2 Key features: Limited range of validity Operating point Mathematical guarantees?

20. Approximation of 2D Surface in 3D Replace a (complicated) surface with a (simpler) function (plane?) Tangent Plane C1 T1 C2 T3 T2 P C3

21. Biological Validity Limited range of interest Buffering Simplifications Time scales Space Reaction speed Vmax 10 5 0 10 20

22. Mathematical Basis: Brook Taylor (1685-1731) Features: Operating point p Derivatives of different orders Factorials Error? For n = 1: linear; n = 2: parabolic, … Problem: polynomials not easy to work with

23. Mathematical Basis (2 or more variables) : Brook Taylor (1685-1731) Features: Operating point (x0, y0) Derivatives of different orders; wrt x, wrt y, mixed Factorials Error? For n = 1: linear; n > 1: messy very soon!

24. A(X, Y) Y A(XOP+X’, YOP+Y’) YOP Y’ X’ XOP X Mathematical Basis (2 or more variables) : Brook Taylor (1685-1731) Recall Approximation of 2D Surface in 3D

25. Possible to Obtain “Less Messy” Nonlinear Approximations? Answer: Power-Law Approximation Derivation: Linear Approximation in logarithmic space F(x) lnF(lnX) linearize exponentiate Result for univariate functions always: F(x) ≈g X f kinetic order rate constant >> Power-law representations are the basis of Biochemical Systems Theory

26. Short-Cut (prove if you like/can) • Each kinetic order (exponent) is equal to: the partial derivative of the approximated function F with respect to the corresponding Xj; multiplied by Xj; and divided by F. • The rate constant is subsequently computed by equating the power-law term and the approximated function F at the operating point. Example: kinetic order rate constant

27. PL Approximation depends on OP

28. Power-Law Approximations of Multivariate Functions Same short-cut! Example: MMRL with explicit enzyme representation: Corresponding power-law representation will have the form

29. Power-Law Approximations of Multivariate Functions (cont’d)

30. Summary of Features of Power-Law Approximations Nonlinear Obtained from linearization in log space (any dimension) Shortcut makes life easy Guaranteed at one operating point of choice Range of validity typically wider than with linear approximation Many interesting properties (e.g., algebraic st-st computation in S-systems) Lots of successful applications

31. X X X X X X X X 1 1 2 2 3 3 4 4 g41 = 0 Recall from Earlier Lecture: Mapping between structure and parameters g41 < 0

32. S-system Form: Alternative Formulations Vi1+ Vi1– Xi Vi,q– Vi,p+

33. Generalized Mass Action Form: Alternative Formulations S-system Form: Vi1+ Vi1– Xi Vi,q– Vi,p+

34. Example of Canonical Model Design

35. Example of Canonical Model Design

36. X1 X2 Recall from Introduction: Limit Cycles Possible (not in linear systems!) (a) (b) a1 = 0.9 or a1 = 1.02 (c) (d) 1.5 3 X2 X1 1.5 0.75 X1 X2 0 0 time time 0 120 240 0 30 60

37. Alternative: Lotka-Volterra Model (Recall bacterial model in earlier lecture) Mass action kinetic (binary interactions) Mass action kinetic, e.g., ai Xi Xi Xi bij ai Xi Xj

38. X X X X X X X X 1 1 2 2 3 3 4 4 Features of Lotka-Volterra Model Very rich behavior Steady state easy to compute: Trivial solutions: Some, all Xi = 0 Nontrivial solutions: Divide by Xi and solve linear system Problems: ? ?

39. Complex Example of Lotka-Volterra Model

40. Other Canonical Models Linear systems Within BST: half-systems (Riccati systems); binary systems Generalized Lotka-Volterra systems (classification theory) Log-lin and Lin-log systems (metabolism) Synergistic and saturable systems (not yet well developed)

41. Summary of Features of Canonical Models Linear or Nonlinear (more complex; typically wider range of validity) Obtained from strict design recipes No true mechanistic basis Excellent default approaches for ill-defined systems Guaranteed at one operating point of choice Model design prescribed Permit specialized methods (e.g., st-st, optimization, parameter estimation), techniques, software Lots of successful applications