NNMCB - Workshop on Systems biology - Cluster … - Workshop on Systems biology K. Sriram Indraprastha Institute of Information Technology [email protected] December 23, 2014 Michaelis-Menten

NNMCB - Workshop on Systems biology

K. Sriram

Indraprastha Institute of Information Technology

[email protected]

December 23, 2014

Michaelis-Menten Kinetics

K. Sriram (IIIT-Delhi) NNMCB December 23, 2014 1 / 27

Learning objectives of this lecture

(i) To build ODE model based on Michaelis-Menten kinetics.

(ii) To understand allostery and cooperativity and build associatedODE model

(iii) To understand molecular switches and build Goldbeter-Koshland(GK) function to generate switches.

(iv) Use GK function to model phosphorylation-dephoshorylation(PdP) reactions that are involved in generating bistability andoscillations through phase plane analysis.


Enzyme properties

Enzymes are catalysts (generally proteins) that help to convert othermolecules called substrates into products, but they themselves are notchanged by the reaction.

Their most important features are catalytic power, specificity andregulation.

Enzymes accelerate the conversion of substrates into products bylowering the energy of activation of the reaction, but have not effecton ∆G or Keq of reaction.

For example, enzymes may aid in converting charge repulsions andallowing reacting molecules to come into contact for the formation ofnew chemical bounds.


Enzyme properties

Or, if the reaction requires breaking of an existing bound, the enzymemay exert a stress on a substrate molecule, rendering a particularbound easily broken.

Enzymes are particularly efficient at speeding up biological reactions,giving increase in speed up to 10 million times or more.

They are also highly specific, usually catalyzing the reaction of onlyone particular substrate or closely related substrates.

Finally, they are typically regulated by an enormously complicated setof positive and negative feedback systems, thus allowing precisecontrol over the rate of reaction.


Enzyme catalysis – example

An example of an enzymatic reaction is the first reaction of theglycolysis, catalyzed by the enzyme hexokinase/


Enzyme catalysis – Derivation of Michaelis-Mentenequation

The reaction scheme can be written as follows (C=complex betweenE and S, namely [ES]):

E + Sk1←−→k−1

Ck2−→ E + P

The equation for the different species follow the mass action and theevolution law is given as:


Michaelis-Menten equation - approximation

In this d [C ]dt = d [E ]

dt = 0

We define Etot the total concentration of enzyme: Etot = E + C =constant


Quasi-steady-state-equation

d [S ]

dt= −k1[E ][S ] + k−1[C ]

d [E ]

dt= −k1[E ][S ] + k−1[C ] + k2[C ] = 0

d [C ]

dt= k1[E ][S ]− k−1C − k2[C ] = 0

d [P]

dt= k2[C ]

The rate of change of total enzyme and complex is

d [E ]

dt= 0,

d [C ]

dt= 0

From this equation, Etot = E + C, we can extract C, and its given as

C =k1Etot [S ]

k1[S ] + (k1 + k2)=

Etot [S ]

[S ] + (k1+k2)k1


Quasi-steady-state equation

When we replace this expression for C in the rate of appearance of P,we obtain:v = d [P]

dt = k2[C] = k2Etot [S]

[S]+(k1+k2)

k1

This is usually written as v = v vmaxS

S+K M

This parameters KM = k1+k2k1

and vmax = k2 ETOT .

The rate is thus similar than in the case of the equilibrium hypothesis(Michaelis-Menten equation); only KM has a slightly differentmeaning.


Lineweaver-Burk plot to estimate parameters

Rewritten in the following manner, the equation gives a straight line,which is useful to determinate the parameters KM and Vmax

(Lineweaver-Burk representation):1v = 1

vmax+ KM

vmax

1S

Zero order reaction is where the enzyme is completely saturated bysubstrate. Increase in substrate do not have any effect on the rates ofthe reaction.

Double‐reciprocal plotDouble reciprocal plot

Zero order

First order


Enzyme inhibition–competitive and non-competitive


Steps involved in deriving competitive inhibition

Write the chemical equations.

Get Etot and perform the SS approximation and Get the rate.

Find the elasticities.

Note: Here, I is the Inhibitor.

Exercise: What will the rate if n moles of the substrate is involved?

i.e. nS + Ek1←−→k−1

E + P.


Allosteric Regulation

The catalytic efficiency of an enzyme depends on the conformation ofits active site.

This conformation depends, in turn, on the overall configuration ofthe protein (its tertiary structure). This configuration, and hence thenature of the active site, can be altered by modifications to thechemical energy landscape of the protein, e.g. the position andstrength of charges.

These modifications can be made by molecules that bind the protein.

This mode of enzyme regulation, called allosteric control, wasproposed by Francois Jacob and Jacques Monod in 1961.


Allosteric Regulation - Different from competitive inhibition

The term allostery (from Greek, allo: other, steros: solid, or shape)emphasizes the distinction from competitive inhibition – the regulatingmolecule need not bear any chemical resemblance to the substrate.

Likewise, the site on the enzyme where the regulator binds (called theallosteric site) can be distinct from the active site in both positionand chemical structure.

Typically, the binding of an allosteric regulator to a protein invokes atransition between a functional state and a non-functional state.

For enzymes, this is typically a transition between a catalyticallyactive form and an inactive form.


Allosteric regulations - reaction and model

Allosteric regulations (Non‐competitive Inhibition)

Consider an enzyme that binds asingle allosteric regulator.Assume that inhibitor has noeffect on substrate binding.

The reactions are

S + EK1←−→k−1

ESk2←→ E + P

E + Iki1←−→k−i1

EI

ES + Ik3←−→k−3

ESI

S + EIki2←−→k−i2

ESI

The rate S → P is given as

vmaxS

(1 + I/KI )(KM + S)

with KI1,I2 = KI .


Allosteric regulations

Allosteric inhibition. Vmax gets affected. This allosteric inhibitor I reducesthe limiting rate Vmax , but doesnot affect the half-saturatingconcentration KM .

More generally, other allostericinhibition schemes impact bothVmax and KM .


Cooperativity

The term cooperativity is used to describe potentially independentbinding events that have a significant influence on one another,leading to nonlinear behaviors.

Cooperativity exhibits sigmoidal behavior unlike the hyperbolicbehavior seen in Michaelis-Menten kinetics.

To address cooperativity, consider the binding of a molecule n ligandsX bind to a protein P simultaneously. The reaction scheme is

P + nxk1←−→k−1

PXn

At equilibrium, Ka = k1k−1

= [PXn][P][x]n , and [Ptotal ] = [P] + [PXn].

The fractional saturation given by Y = [PXn][P]total

= xn1ka+xn

=xn

Kd + xn


Sigmoidal plot of Hill’s equation

Reaction rate


Molecular switch in PdP and GTP reactions


Example of hyperbolic response -phosphorylation-dephosphorylation (PdP) reactions

Signal ‐ Response curve Phase plot


Ultrasensitivity and Zero-order ultrasensitivity

Definition of ultrasensitivity: A modest change in theconcentration of substrate of an enzyme, or a modest change in theactivity of an enzyme, greatly changes the net rate of the reactioncatalyzed by that enzyme.

Zero-order ultrasensitivity : Two enzymes catalyze the samebiochemical transformation, one in the forward direction and theother in the reverse direction.

Both enzymes must be nearly saturated with substrate, so that boththe reverse and forward reactions are approximately zero-order.

At steady state, the substrate concentrations are often such that therates of both reactions are nearly equal.


Zero-order ultrasensitivity -phosphorylation-dephosphorylation (PdP) reactions

Kinase

Switch‐like

Phosphatase behavior

Now, suppose there is a small increase in the activity (Vmax)of the enzyme catalyzing the forward reaction. This unbalances the rates

h h f d i i fso that the forward reaction is faster.

Since this enzyme is saturated with substrate, a large drop in the y g pconcentration of substrate is required to bring the rates back into balance.

Thus, a small change in enzyme activity results in a large change in the ratio of substrate to product.


Zero-order ultrasensitivity - Model

Kinase

At steady Phosphatase

(Conservation relation)

ystate

simplifying and scaling the relevant variables,

Solving this quadratic equation,

(i) CHECK ! (ii) How this

expression is obtained?


How to determine the sigmoidal nature of GK function?

Consider the scaled steady-state equation

u1x(J2 + 1− x) = u2(1− x)(J1 + x)

It is easier to think of u1 as a function of x than x as a function of u1.

So u1 = u2J1+x

J2+1−x1−xx

Zeros of u1 are x = and x = - J1.

Vertical asymptotes are x = 1 + J2 and x = 0.

For 0 < J1, J2 << 1, The curve should exhibit switch-like behavior.


Phase-plane of PdP reactions with negative feedback

NFL

degradation

Shape of the nullclinesStable‐Focus

The X‐nullcline is a hyperbola

Shape of the nullclines

YP ‐nullcline is a sigmoidal curve

Nature of dynamicsHere the negative feedback brings about homeostasis.

The fixed point (FP) here it is globally stable.


Phase-plane of PdP reactions with positive feedback

PFL

Shape of the nullclines

Nature of dynamics Notice that as we increase or decrease S, the X‐nullcline moves down or up.

Yp nullcline FP3

Nature of dynamics

Bistability

There is a range of S values, S ε (Sc1, Sc2), for which the nullclines intersect in

Yp‐nullcline FP3

three places.

The points at the end of this range, FP2

where the system changes from one to three steady states, are called saddle‐node (SN) bifurcation points.

FP1


Networks that oscillate(ii) Substrate‐Depletion

(i) Activator‐Inhibitor

Example of Substrate‐DepletionActivate

p p

(iii) DelayedNegative Feedback ODE Model of DNF

degradationdegradation

Build an ODE model for Activator-Inhibitor (i) andSubstrate-depletion (ii) modules.

Simulate delayed negative feedback (iii) using MATLAB and analyzeits dynamics. The model for DNF is given.


Documents

NNMCB - Workshop on Systems biology - Cluster … - Workshop on Systems biology K. Sriram Indraprastha Institute of Information Technology [email protected] December 23, 2014 Michaelis-Menten