1. BENG/CHEM/Pharm/MATH 276 HHMI Interfaces Lab 2: Numerical Analysis for Multi-Scale Biology Modeling Cell Biochemical and Biophysical Networks Britton Boras

2. Cell Systems Models Systems of ordinary differential and algebraic equations – Conservation principles – mass, charge, energy – Empirical relations – State-transition models – Lumped parameters and compartments (“common pool” models) – Computationally tractable but often data limited Examples – Biochemical reaction networks Signaling pathways Metabolic networks Gene Regulatory networks – Chemical buffering and transport – Transmembrane ionic currents – Protein and complex state transitions

3. Biochemical Networks Binding Reactions – Receptor-ligand – Complex formation – Inhibition Transformation – Enzymatic conversion – Degradation – Michaelis-Menten Enzyme Reaction scheme – Quasi-equilibrium assumption (k 1, k -1 >> k 2 ) preferable to quasi- steady state assumption ([S] >> [E] + [ES]) Compartmentation – Organelles and subspaces – Scaffolds and complexes Conservation relations – Algebraic equations, e.g. Complexation coefficient  =  free /  cmplx [X] effective =  [X]

4. Saucerman, JJ et al., J Biol Chem 278: 47997 (2003) Integrated Example:  β -Adrenergic Regulation of Excitation- Contraction Coupling

5. Whole Cell Model of Cardiac Myocyte Activation

6. Oxygen Binding Hemoglobin

7. Model Equations Governing ordinary differential equations (ODEs) come from mass conversation:

8. Virtual Cell

9. What is Virtual Cell Virtual cell is subcellular based modeling program Developed by the National Resource for Cell Analysis and Modeling (NRCAM) and the University of Connecticut Health Center Creates a link between experimental and computational models Provides a graphical interface for definition of pathways and geometries Exports a mathematical description that can be read by other solvers

10. Advantages to Virtual Cell PKI = p(32)*p(36)/(p(36)+y(17)+y(18)); A2RC_I = (y(17)/p(35))*y(17)*(1+PKI/p(36)); A2R_I = y(17)*(1+PKI/p(36)); A2RC_II = (y(18)/p(35))*y(18)*(1+PKI/p(36)); A2R_II = y(18)*(1+PKI/p(36)); ARC_I = (p(33)/y(16))*A2RC_I; ARC_II = (p(33)/y(16))*A2RC_II; ydot(16) = y(15)-(ARC_I+2*A2RC_I+2*A2R_I)- (ARC_II+2*A2RC_II+2*A2R_II)-y(16); PKAtemp = p(33)*p(34)/p(35)+p(33)*y(16)/p(35)+y(16)^2/p(35); ydot(17) = 2*p(30)*y(16)^2- y(17)*(1+PKI/p(36))*(PKAtemp*y(17)+y(16)^2); ydot(18) = 2*p(31)*y(16)^2- y(18)*(1+PKI/p(36))*(PKAtemp*y(18)+y(16)^2); Virtual Cell Model Matlab Model

11. Defining a Biomodel

12. Defining the Biomodel Each reaction is defined as a function of its flux

13. Creating the Mathematical Model Mathematical model contains initial conditions, reaction kinetics, volume domains, and electrical mapping (if present) Non-spatial biomodels can be exported as either VCML, XML, or Matlab files

14. Create the systems diagram Go to Reaction Diagram Use Species Tool (green Dot) to create 6 species Name the species as shown above Use the Select tool (white arrow) to move the species away from each other

15. Creating the systems diagram Click the RX Connection Tool then left click and hold on Hemo_0 and drag to Hemo_1 and release. Click and hold on the O2 species and drag to the reaction box Repeat this process connecting each out hemoglobin species in order with O2

16. Defining the Reactions Use the Select Tool (white arrow) click the first reaction and change the forward rate to 40 1/(uM*s) and Kr to 1000 1/s. Do the same with each of the other reactions

17. Defining the Experiment Right Click Applications-Add new, Deterministic Click Specifications Clamp O2 concentration at 40 uM and set hemo 40 uM.

18. Initial Conditions

19. Creating the Simulation Click Simulation and then new simulation. Change the End Time from 1 sec to 0.1 sec Click Edit Simulation Scan O2 concentration from 0.2 to 200 uM on a log scale with 15 values.

20. Defining a New Function Click Output Functions and define a new function for uM of oxygen bound Return to the Simulations tab, highlight the simulation, and click run. (it will ask you to save first)

21. How is Virtual Models Solved The simulations are run over the internet on 84 servers with 256 GHz total CPU power and 119 GB total RAM. Currently the storage capacity is 11.7 Tb. Only a client is on the user’s computer (it does not matter how powerful your computer is) Blends a defined geometry with a defined biochemical schematic

22. Viewing the Results Click Simulation Results to review the results of the model

23. Results Tab Look at various species at different O 2 concentrations with time. Open Excel and make a graph of steady state BoundO2 function with increasing O 2 concentration

24. Extension: Go back to the reaction diagram and change the forward rates to: – R1=80, R2=80, R3=80, R4=80 – R1=10, R2=70, R3=140, R4=200 – R1=200, R2=140, R3=70, R4=10 Graph the results in Excel What do these graphs show about Cooperativity vs. K D ?