Chapter 8 Applications In physics In biology In chemistry In engineering In political sciences In social sciences In business

In chemistry – chemical reaction Biochemical reactions: –Take place in all living organisms –Most of them involve proteins called as enzymes –Enzymes react selectively on compounds: substrates –Biological & biochemical processes are complicated –Develop a simplifying model to understand the phenomena –Give qualitative understanding –First step to develop more realistic & complicated model

Reaction kinetics Basic enzyme reaction: Michaelis & Menten (1913) –A substrate S reacting with an enzyme E to form a complex SE –The complex SE is converted into a product P and the enzyme The laws of mass action: the rate of a reaction is proportional to the product of the concentrations of the reactants

Reaction equations Concentrations of the reactants: Nonlinear reaction equations Explain: The first equation for s is simply the statement that the rate of change of the concentration [S] is made up of a loss rate proportional to [S] [E] and a gain rate proportional to [SE].

Model reduction Initial conditions: The last equation is uncoupled Conservation of enzyme: catalyst Reduced system

Pseudo-steady state Pseudo-steady state solution: The reaction of complex to product is much faster than that of substrate to complex, i.e. enzyme is almost at equilibrium The equation

Pseudo-steady state solution Define Michaelis constant Pseudo-steady state solution Determine rate of reaction v –Take a sequence of different initial values of –Measure the corresponding variation of s with t, –Rate of reaction –Obtain for each experiment a measurement of the initial rate

Lineweaver-Burk plot

Michaelis-Menten rate

Qualitative analysis Nondimensionalization: Dimensionless equation Qualitative understanding –Steady state: u=v=0 –v increases from v=0 until attains its maximum at v=u/(u+K) then decreases to v=0 –u decreases monotonically from u=1 to u=0

Qualitative analysis

Michaelis-Menhten theory Pseudo-steady state hypothesis: The remarkable catalytic effectiveness of enzymes is reflected in the small concentrations needed in their reactions as compared with the concentrations of the substrates. Approximate (asymptotic) solution: –Assume –Substitute and equate powers of –The O(1) equations:

Michaelis-Menten theory Solution: (nonsingular or outer solution, valid for ) Difficulty: The second equation is algebraic & does not satisfy the initial condition

Michaelis-Menten theory The solution is not a uniformly valid approximation for all The original problem is a singular perturbation problem since the second equation is multiplied by a small parameter The assumption is not valid near Initial layer exists Introduce the transformations New equations

Michaelis-Menten theory Assume O(1) equations The solutions (singular or inner solution, valid for )

Michaelis-Menten theory Matching: –The limit of the outer solution when –The limit of the inner solution when Initial (or boundary) layer:

Michaelis-Menten theory Singular perturbation, a systemic way –Outer solution in the form of a regular series expansion –Inner solution expansion

Michaelis-Menten theory –Initial conditions: –Thus the singular solutions are determined completely –Outer solutions –Matching of the inner and outer solutions

Michaelis-Menten theory Uniformly expansion

Michaelis-Menten theory

Uniformly matched asymptotic expansion: inner+outer-middle Explain –Rapid change in substrate-enzyme takes place in dimensionless time –Very short, in many experimental cases, singular solutions is not observed –The reaction for the complex is essentially in a steady state –The v-reaction is so fast it is more or less in equilibrium at all times –This is Michaelis and Menten’s pseudo-steady state hypothesis

Other chemical reactions Cooperative phenomena –Enzyme has more than one binding site for substrate molecules –An enzyme + a substrate is called as cooperative: if a single enzyme molecule, after binding a substrate molecule at one site can then bind another substrate molecule at another site. –Example: enzyme molecule E binds a substrate molecule S to form a single bound substrate-enzyme complex C1. C1 can break to form a product P and enzyme E & combine with another substrate molecule S to form a dual bound substrate-enzyme complex C2. C2 breaks down to form the product P and single bound complex C1. Autocatalysis, Activation & Inhibition: systems with feedback controls

Other models Biological oscillators & switches: Feedback control Reaction diffusion, Chemotaxis