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 (kf and kd) may depend on environmental factors such as temperature or pressure

Enzymes or catalysts can also affect kf and kd

]BA[k]B[]A[kdt

]BA[d

BABA2

2d2

f2

2

k

k

f

d

Chemical Reactions

Can make assumptions to simplify equations Abundant species: suppose [A] >> [B] and [A2B],

then [A] will be relatively constant throughout reaction Stoichiometry: when [B] increases, [A2B] must

decrease and vice versa, so [B] + [A2B] = constant

Let [A] = CA and [B] + [A2B] = Ct

]BA)[kk(CCkdt

]BA[d

]BA[k])BA[C(Ckdt

]BA[d

2dft2

Af2

2d2t2

Af2

Hemoglobin and Oxygen

Hemoglobin molecules carry O2 in blood

Each molecule has 4 heme groups Each heme group binds to one O2 molecule

Binding of O2 molecule changes affinity for next O2 molecule

Changes in binding affinity are represented by different rate constants at each step

84

k

k264

k

k244

k

k224

k

k24 OHbOOHbOOHbOOHbOHb

4f

4d

3f

3d

2f

2d

1f

1d

Hemoglobin Equations

]OHb[k]O][OHb[kdt

]OHb[d

]O][OHb[k]OHb[k]OHb[k]O][OHb[kdt

]OHb[d

]O][OHb[k]OHb[k]OHb[k]O][OHb[kdt

]OHb[d

]O][OHb[k]OHb[k]OHb[k]O][Hb[kdt

]OHb[d

]O][Hb[k]OHb[kdt

]Hb[d

OHbOOHbOOHbOOHbOHb

844d2644f84

2644f643d844d2443f64

2443f442d643d2242f44

2242f241d442d241f24

241f241d4

84

k

k264

k

k244

k

k224

k

k24

4f

4d

3f

3d

2f

2d

1f

1d

Hemoglobin Equations

Use state variables– x0 = [Hb4]– x1 = [Hb4O2]– x2 = [Hb4O4]– x3 = [Hb4O6]– x4 = [Hb4O8]

Gases in solution are represented by their partial pressure – PO2 = [O2]

Can also substitute for sum of Hb4 molecules– sum(xi) = constant

44d234f4

234f33d44d223f3

223f22d33d212f2

212f11d22d201f1

201f11d0

xkPOxkdt

dx

POxkxkxkPOxkdt

dx

POxkxkxkPOxkdt

dx

POxkxkxkPOxkdt

dx

POxkxkdt

dx

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(C2 – C1)

Analogous to flow of fluid in pipe due to pressure difference: Q = k(P2 – P1)

The proportionality constant k is determined by the resistance to particle flow – cross section area, distance between C1 and C2, how many other particles in fluid, etc.

Diffusion

The flow of particles can alter the concentration gradient

Q = k(C2 – C1) represents the flow of particles from C2 to C1

dC1/dt & dC2/dt are proportional to Q– Q in units of particles / time– C1 & C2 in units of particles / volume– dC1/dt & dC2/dt in units of particles / volume / time– Proportionality constant has units of 1 / volume

Can rewrite equation as dC1/dt = -dC2/dt = k(C2 – C1)– 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

Vout/Vin = -R2/R1

– 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:

dt)t(u)t(y

)t(x)t(y

)t(u)t(xdt

d

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

)t(udt

d)t(y