Presentation Schedule

Homework 8• Compare the tumor-immune model using Von

Bertalanffy growth to the one presented in class using a qualitative analysis… Determine the number of steady states and their stability using qualitative approaches when analytical progress is not possible. Be sure to nondimensionalize and generate phase portraits. Does the qualitative behavior differ from that of the model presented in class? If so, how?

• Consider sigma to be your bifurcation parameter

Models for Molecular Events

Recetor-Ligand Binding

Definitions• Receptor

– a protein molecule, embedded in either the plasma membrane or cytoplasm of a cell, to which a mobile signaling (or "signal") molecule may attach.

• Ligand– a signal triggering substance that is able to bind

to and form a complex with a biomolecule to serve a biological purpose

– ay be a peptide (such as a neurotransmitter), a hormone, a pharmaceutical drug, or a toxin,

Why Receptor Ligand Binding is Important

• Individual cells must be able to interact with a complex variety of molecules, derived from not only the outside environment but also generated within the cell itself. Protein-ligand binding has an important role in the function of living organisms and is one method that the cell uses to interact with these wide variety of molecules. When such binding occurs, the receptor undergoes conformational changes, which ordinarily initiates a cellular response.

Example: VEGF Receptors

Possible Cellular Responses

Cellular Uptake

• Molecules are taken up by cells in different ways– Glucose is transported inside cells by

facilitated diffusion– Other water soluble molecules must be carried

into the cell via receptor-mediated endocytosis• process by which cells internalize molecules via the

inward budding of plasma membrane vesicles containing proteins with receptor sites specific to the molecules being internalized.

Facilitated Diffusion and Receptor-Mediated

Endocytosis

Chemostat Revisited• Recall that we modeled

nutrient uptake using

• Why should saturation occur?– Limited number of receptors

or carrier molecules– Limited rate of

internalization and release €

k(C) =rC

a+C

C

r/2

r

a

€

k(C) =rC

a+C

Goal• Model the process of nutrient uptake• Schematic Diagram

STEP 1• Reaction Diagram

• Reaction diagrams can be converted to a system of odes that describe the rates of change of the concentration of the reactants

+

Extracellular Nutrient

Free Recepto

r

N-R Complex

+

Intracellular Nutrient

k2

N R C RP

+ k1

k-1

+

The Law of Mass Action

• To go from molecules to concentration we use the Law of Mass Action– When two or more reactants are involved

in a reaction step, the rate of the reaction is proportional to the product of the concentrations of the reactants.

– Convention: ki’s are the proportionality constants

Model Variables

Variable Definition

Units

r [R] #/cell

n [N] Moles/volume

c [C] #/cell

p [P] moles/volume

The Model Equations

€

dr

dt= −k1rn + k−1c + k2c

€

dn

dt= −k1rn + k−1c

€

dc

dt= k1rn − k−1c − k2c

€

dp

dt= k2c

€

r(0) = r0

€

n(0) = n0

€

c(0) = 0

€

p(0) = 0

Notes

• The p equation is decoupled– We only need to consider 3 equations

• The total number of receptors is conserved

– We only need to consider 2 differential equations (the c and n equations), together with:

€

dr

dt+dc

dt= 0

€

r + c = r0

€

r = r0 − c

Reduced Model

• Note: Because this is a system of two equations, we use the traditional stability and phase plane analysis, but let’s do something different first.

€

dn

dt= −k1r0n + k1n + k−1[ ]c

€

dc

dt= k1r0n − k1n + k−1 + k2[ ]c

€

n(0) = n0

€

c(0) = 0

Quasi-Steady State Assumption

• The concentration of the substrate-bound enzyme (and hence also the unbound enzyme) change much more slowly than those of the product and substrate.

• Rationale– Small molecules like glucose are found in

higher concentrations than the receptors are– If this is true, then receptors are working at

maximal capacity• Therefore the occupancy rate is virtually constant

Quasi-Steady State Approximation

• The QSSA is written as

€

dc

dt≈ 0

€

0 ≈ k1r0n − k1n + k−1 + k2[ ]c

€

c ≈k1r0n

k1n + k−1 + k2

Michaelis-Menton Kinetics

• A simple substitution shows that we have derived the Michaelis-Mention kinetic form that we used in the chemostat model.

€

dn

dt= −

kmaxn

kn + n

€

kmax = k2r0

€

kn =k−1 + k2k1

Problem with QSSA

• By assuming that dc/dt = 0, we changed the nature of the model from 2 ODEs to one ODE and one algebraic expression. There must be consequences for doing this.

• To see which timescales QSSA is valid on, let’s nondimensionalize.

Nondimensionalization

• The nutrient and complexes are scaled by their initial conditions. Time is scaled by receptor density multiplied by the association rate.

€

u =n

n0

€

τ = 1

k1r0t

€

v =c

r0

€

ε = r0n0

€

k =k−1 + k2k1n0

€

λ =k2k1n0

Nondimensional Equations

• Now we see that assuming dc/dt = 0 is equivalent to assuming that ε << 1, which means r0 << n0.

€

du

dt= −u+ K − λ + u( )v

€

ε dvdt= u− K + u( )v

€

u(0) =1

€

v(0) = 0

Validity of QSSA• So, on timescales of the order

1/(k1r0) (ie long timescales), receptor-mediated nutrient uptake can to approximated by

€

0 = u− (K + u)v

€

v =u

K + u€

du

dt= −u+ K − λ + u( )v

€

du

dt=−λu

K + u

Behavior of Solutions• u is a decreasing function of time and v

decreases if u decreases.• Therefore, on this timescale ( = long

times), both the nutrient and complex concentrations are decreasing

• This can’t always be true, recall that we started with c(0) = 0.

• Let’s see how the solutions behave on short timescales

Nondimensionalize

• The nutrient and complexes are scaled by their initial conditions. Time is scaled by nutrient concentration multiplied by the association rate.

€

u =n

n0

€

τ = 1

k1n0t

€

v =c

r0

€

ε = r0n0

€

k =k−1 + k2k1n0

€

λ =k2k1n0

On Short Timescales

• We can now predict how receptors fill up

€

du

dt= −εu+ ε K − λ + u( )v

€

dv

dt= u− K + u( )v

€

u(0) =1

€

v(0) = 0

On Short Timescales• Now if ε = r0/n0 ~ 0, we have

• We can now predict how receptors fill up

€

du

dt= 0

€

dv

dt= u− K + u( )v

€

u(0) =1

€

v(0) = 0

Short Timescale Solutions

So v rises quickly to a maximum on short timescales.

€

u =1

€

dv

dt=1− K +1( )v

€

v =1− e−(K +1)τ

K +1

Complete Behavior

• Initially, v rapidly rises which means receptor density quickly increases

• Eventually, the nutrient is depleted and the the density of bound complexes follows suit

• The behavior of the system can be completely determined by solving approximate equations on two different timescales

QSSA vs Full Model Behavior

QSSA vs Full Model Behavior

Traditional Analysis

• The only steady state is u = 0, v = 0 and it is stable

• Eventually all of the nutrient is consumed and internalized and all of the receptors are empty