Modeling Chemical Systems BIOE 4200

Chemical Systems Every physiologic systems depends on multiple chemical reactions Examples include hormone-receptor interactions, enzyme-substrate binding, and blood gas dynamics The diffusion of reactants or products within tissues affects this process Chemical reaction models may be needed depending on the level of detail

Chemical Reactions Differential equations can be derived directly from reaction stoichiometry Forward and reverse reaction rates (k f and k d ) may depend on environmental factors such as temperature or pressure Enzymes or catalysts can also affect k f and k d

Chemical Reactions Can make assumptions to simplify equations Abundant species: suppose [A] >> [B] and [A 2 B], then [A] will be relatively constant throughout reaction Stoichiometry: when [B] increases, [A 2 B] must decrease and vice versa, so [B] + [A 2 B] = constant Let [A] = C A and [B] + [A 2 B] = C t

Hemoglobin and Oxygen Hemoglobin molecules carry O 2 in blood Each molecule has 4 heme groups Each heme group binds to one O 2 molecule Binding of O 2 molecule changes affinity for next O 2 molecule Changes in binding affinity are represented by different rate constants at each step

Hemoglobin Equations

Use state variables – x 0 = [Hb 4 ] – x 1 = [Hb 4 O 2 ] – x 2 = [Hb 4 O 4 ] – x 3 = [Hb 4 O 6 ] – x 4 = [Hb 4 O 8 ] Gases in solution are represented by their partial pressure – PO 2 = [O 2 ] Can also substitute for sum of Hb 4 molecules – sum(x i ) = constant

Diffusion Particles will flow between fluid regions that have different particle concentration Concentration difference between two points can serve as a “driving force” for particle flow Flow is proportional to concentration difference: Q = k(C 2 – C 1 ) Analogous to flow of fluid in pipe due to pressure difference: Q = k(P 2 – P 1 ) The proportionality constant k is determined by the resistance to particle flow – cross section area, distance between C 1 and C 2, how many other particles in fluid, etc.

Diffusion The flow of particles can alter the concentration gradient Q = k(C 2 – C 1 ) represents the flow of particles from C 2 to C 1 dC 1 /dt & dC 2 /dt are proportional to Q – Q in units of particles / time – C 1 & C 2 in units of particles / volume – dC 1 /dt & dC 2 /dt in units of particles / volume / time – Proportionality constant has units of 1 / volume Can rewrite equation as dC 1 /dt = -dC 2 /dt = k(C 2 – C 1 ) – Proportionality constant has units of 1 / time

Scaling and Gain Some processes directly transform input to output (y = ku) No state equation is needed Useful to model electrical amplification Example: op-amp in inverted configuration has gain V out /V in = -R 2 /R 1 – Scaling factor k has no units Useful for transforming input units to output units Example: digital speedometer measures MPH, output is V – Scaling factor k has units V/MPH

Integration Some processes are needed to transform a rate (input) to an absolute amount (output) – Convert from mg/sec to mg – Convert from velocity to displacement Original equation is Can represent this using state equations:

Differentiation Some processes are needed to transform an absolute amount (input) to a rate (output) – Convert from mg to mg/sec – Convert from displacement to velocity Original equation is Cannot represent this using state equations! Will be able to represent this using transfer functions (later in course) Not a big deal – don’t usually need to convert variables this way