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Modeling biochemical reactions
Matthew Macauley
Department of Mathematical Sciences
Clemson University
http://www.math.clemson.edu/~macaule/
Math 4500, Spring 2016
M. Macauley (Clemson)
Modeling biochemical reactions
Math 4500, Spring 2016
1 / 10
Overview
In biochemistry, 2+ species, or “reactants” can react if they come toegether and
collide.
Alternatively, one species can degrade.
More is needed, though: correct orientation, enough energy, etc.
Examples
CH4
2O2
ÝÑ CO2
2H2 O
(burning of methane)
OH ÝÑ H2 O
unfolded protein ÝÑ folded protein
2SO2 O2 Ý
âÝÝá
Ý 2SO3
H
ÝÑ O2 O
2O3 ÝÑ 3O2
O3
M. Macauley (Clemson)
Modeling biochemical reactions
Math 4500, Spring 2016
2 / 10
Mass-action kinetics
Classification of reactions:
A ÝÑ P:
A
B
A
B
“uni-molecular”
ÝÑ P: “bi-molecular”
C ÝÑ P: “tri-molecular”
Law of mass-action kinetics
A reaction rate is proportional to the probability of collision of reactants involved.
Assume this probability is proportional to the concentration of each reactant R,
denoted rR s.
ODE model
A ÝÑ P:
k
A
B
k
ÝÑ
P:
A
B
k
Ý
âá
Ý P:
k
1
2
M. Macauley (Clemson)
d rP s
k rAs
dt
d rP s
k rAsrB s
dt
d rP s
k1 rAsrB s k2 rP s
dt
Modeling biochemical reactions
Math 4500, Spring 2016
3 / 10
Mass-action kinetics
Enzymes are proteins that catalyze reactions (up to 1012 -fold!)
An example
Consider the following chemical reaction
E
S
k
k
Ý
E
âá
Ý ES ÝÑ
k
1
3
P
2
E
enzyme, S substrate, ES enzyme-substrate complex, and P product.
$ d rES s
'
k1 rE srS s pk2 k3 qrES s
'
& dt
' ddtrP s k rES s
'
%E rE s rES s,
3
0
E0
initial enzyme concentration
Assumptions
E0 is constant.
Enzyme-substrate complex reaches equilibrium much earlier than the product
d rES s
does, so
0.
dt
M. Macauley (Clemson)
Modeling biochemical reactions
Math 4500, Spring 2016
4 / 10
Mass-action kinetics
Goal
Write the differential equation
Since
d rES s
dt
d rP s
dt
k3 rES s in terms of rS s, not rES s.
0, we can simplify the ODE for rES s:
d rES s
k1 rE srS s pk2 k3 qrES s 0 .
dt
Upon solving for rE s, we get
rE s pk2 k1krS3 qrs ES s .
Plugging this into E0 rE s rES s and solving for rES s:
rES s k Ek 0 rS s .
rS s
k
2
3
1
Alas, we can write
d rP s
dt
M. Macauley (Clemson)
k3 rES s k2
k3 E0 rS s
k3
rS s
k1
KVmmax rrSSss .
Modeling biochemical reactions
Math 4500, Spring 2016
5 / 10
Michaelis–Menten equation
Recall the following chemical reaction:
E
S
k
Ýká
E
â
Ý ES ÝÑ
k
1
3
P
2
E
enzyme, S substrate, ES enzyme-substrate complex, and P product.
Definition
The Michaelis–Menten equation is one of the best-known models of enzyme kinetics.
d rP s
dt
KVmmax rrSSss ,
loooomoooon
f
where Vmax
k3 E0 ,
and Km
k2 k1 k3
prS sq
Remarks
The “reaction rate”, f prS sq, is a strictly increasing function of rS s.
lim f prS sq Vmax ,
rS sÑ8
(biologically, the maximum reaction rate)
f pKm q 12 Vmax .
The reaction rate f prS sq is proportional to E0 .
M. Macauley (Clemson)
Modeling biochemical reactions
Math 4500, Spring 2016
6 / 10
Michaelis–Menten equation
Recall the following chemical reaction:
E
S
k
k
Ý
E
âá
Ý ES ÝÑ
k
2
3
P
1
E
enzyme, S substrate, ES enzyme-substrate complex, and P product.
Further assumptions
Substrate concentration is conserved: S0
E0
! S0 , so rES s ! rS s and rP s.
Together, this means S0
rS s rES s rP s.
rS s rP s. Taking dtd of both sides yields
rP s Vmax rS s .
d rS s
ddt
km r S s
dt
Usually, Vmax , Km , and S0 are known quanities. This is now something we can easily
solve, graph, analyze, etc.
M. Macauley (Clemson)
Modeling biochemical reactions
Math 4500, Spring 2016
7 / 10
Multi-molecule binding
Consider a reaction where n molecules of a substrate S react with an enzyme E :
E
nS
k
k
Ý
E
âá
Ý ESn ÝÑ
k
1
3
P
2
The enzyme-substrate complex here is ESn . By mass-action kinetics,
$ d rES s
'
k rE srS s pk
'
& dt
' ddtrP s k rES s
'
%
n
n
1
3
2
k3 qrESn s
n
rE s rESn s, E0 initial enzyme concentration
As before, assume rESn s reaches equilibrium much quicker than rP s and rS s:
d rESn s
0 ùñ rE s pk2 k1krS3 qrsnESn s .
dt
Plugging this into E0 rE s rESn s and solving for rESn s yields
n
n
d rP s
ùñ
KVmmax rrSSssn .
rESn s k Ek 0 rS s n
dt
rS s
k
E0
2
3
1
M. Macauley (Clemson)
Modeling biochemical reactions
Math 4500, Spring 2016
8 / 10
Multi-molecule binding
Hill equation
Given the chemical reaction
k
Ýká
E P
â
Ý ESn ÝÑ
k
we derived the following ODE involving rP s and rS s:
d rP s
Vmax rS sn
k2 k3
,
where Vmax k3 E0 , and Km n
dt
K
r
S
s
k1
m
looooomooooon
E
nS
2
3
1
f
prS sq
This is called the Hill equation with Hill coefficient n.
Remarks
The “reaction rate”, f prS sq, is a strictly increasing function of rS s.
lim f prS sq V
max
rS sÑ8
1{n
f pKm q 12 Vmax .
,
(biologically, the maximum reaction rate)
The reaction rate f prS sq is proportional to E0 .
n
1 is just the Michaelis–Menden equation.
M. Macauley (Clemson)
Modeling biochemical reactions
Math 4500, Spring 2016
9 / 10
Hill equations
The following shows several “Hill functions” y
M. Macauley (Clemson)
1t
Modeling biochemical reactions
n
tn
, for various values of n.
Math 4500, Spring 2016
10 / 10