10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. K. Dane Wittrup
Lecture 3: Kinetics of Cell Growth and Enzymes
This lecture covers: cell growth kinetics, substrate uptake and product formation in
microbial growth, enzyme kinetics, and the Michaelis-Menten rate form.
Biological Rate Laws- Enzymes and Cell Growth
Rate Law: − rA = f (C A ,CB ,CP ,T , pH ,...)
Why would you need a rate law?
-Predictive description of a production process
-Design tool for forming a desired product
-Consistency or inconsistency with alternative mechanism
For a hypothesized mechanism, we can often derive an exact, closed form analytical
solution.
If not, it’s used to approximate:
-some reactions go rapidly to equilibrium
-the concentrations of some species rapidly reach their steady state values
-the rate-limiting step
Or, as a last resort- numerical solution
X=first order
-rA
X
X
− rA = kC A
=second order
−rA = kC A 2
=zero order
− rA = k
X
X
X
CA
Figure 1. Rate versus concentration graphs for zero, first, and second order
reactions.
Enzymes-Biological Catalysts
S=Substrate
E=Enzyme
Cite as: K. Dane Wittrup, course materials for 10.37 Chemical and Biological Reaction Engineering, Spring
2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on
[DD Month YYYY].
Energy
Activation energy
decreases with E
(enzyme)
S
P
Reaction Progress
Figure 2. An energy diagram for a reaction with and without an enzyme. The
activation energy is lower with an enzyme, so the reaction proceeds faster.
CS
Initial Rate
− rS = −
dCS
dt
t =0
Time
Figure 3. The slope of the concentration versus time curve at time = 0 can give the
initial rate.
Michaelis Menten
~1st order
~0th order
-rS
X
X
X
X
X
X
X
X X X
X
What gives change
from 1st order to 0th
order?
CS
Figure 4. Michaelis-Menten kinetics.
Hypothesized Mechanism:
-Encounter complex ES formed
-Irreversible reaction occurs
In more complex cases,
-Rapid release of product from complex
these are not always true
-Assume rapid equilibrium is reached in the formation of ES
10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. K. Dane Wittrup
Lecture 3
Page 2 of 6
Cite as: K. Dane Wittrup, course materials for 10.37 Chemical and Biological Reaction Engineering, Spring
2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on
[DD Month YYYY].
Later, Briggs-Haldane derived a law with steady state assumption for ES and they
got the same rate law as Michaelis-Menten. However, the Briggs-Haldane method is
more generally applicable.
1
ZZZ
X
E + S YZZ
Z ES
k
k
−1
ES ⎯⎯→ E + P
kcat
Steady state assumption on ES
→
dCES
=0
dt
Material Balance on ES:
dCES
= k1CE CS − k−1CES − kcat CES ≈ 0 (steady state)
dt
Free enzyme concentration ≠ CE ,0
because CE ,0
CE ,0 = CE + CES
CS ,0
0 = k1 (CE ,0 − CES )CS − CES (k−1 + kcat )
Cs ≈ CS ,0
Solve for CES (steady state solution)
CES =
CE ,0CS
k−1 + kcat
+ CS
k1
− rS = rp =
dC p
dt
= kcat CES =
kcat CE ,0CS
V C
= max S
k−1 + kcat
+ C S K m + CS
k1
Vmax = kcat CE ,0 = maximum reaction velocity
Km =
k−1 + kcat
= Michaelis constant
k1
This model fits the data:
Vmax
CS
Km
-As CS K m , − rS → Vmax
-As CS K m , − rS →
10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. K. Dane Wittrup
Lecture 3
Page 3 of 6
Cite as: K. Dane Wittrup, course materials for 10.37 Chemical and Biological Reaction Engineering, Spring
2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on
[DD Month YYYY].
-rS
X
X
X
X
X X X
X
The rate flattens out
because all of the
enzyme is bound.
X
X
CS
Figure 5. Michaelis-Menten kinetics.
-This rate law reappears in heterogeneous catalysis (Langmuir-Hinshelwood)
-In the actual Michaelis-Menten derivation Kmax was an equilibrium constant
-Briggs and Haldane showed that this rate law still works for steady state
-Steady state approximation is more general
-Steady state approximation is consistent with the data
-Steady state is defined over a limited period of time. Because there is an
irreversible step, eventually all of the intermediate will be consumed.
-Equilibrium assumption is not exactly correct. The irreversible step prevents
true equilibrium.
Growth Kinetics
A population of single cells, growing without limitations
N=#cells per volume
dN
= μ N , μ = constant, specific growth rate
dt
N = N 0e μ t − exponential growth while true, "Malthusian growth"
Growth Stops
-Run out of nutrients
-Accumulate toxic byproducts
Nutrient Effect on μ (Monod)
μ C
μ = max S
K S + CS
This expression is purely empirical (no mechanistic meaning). Even the fit is not that
good.
10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. K. Dane Wittrup
Lecture 3
Page 4 of 6
Cite as: K. Dane Wittrup, course materials for 10.37 Chemical and Biological Reaction Engineering, Spring
2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on
[DD Month YYYY].
μ C
dN
= max S N
dt K S + CS
We will return to this when we
look at bioreactors
dCS
−1
=
μN
dt
Yx s
I=inhibitor or toxin
μ ∝ e − kC
μ
I
μ∝
1
(1 − kCI )
μ∝
kI
k I + CI
All forms fit the data
CI
Figure 6. Growth rate versus concentration of inhibitor. Many functional forms fit
the curve.
CS=concentration of growth limiting
substrate, often glucose
μ
X
X
X
X
X
X
X X X
X
KS
CS
Figure 7. Growth rate versus concentration of growth limiting substrate.
10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. K. Dane Wittrup
Lecture 3
Page 5 of 6
Cite as: K. Dane Wittrup, course materials for 10.37 Chemical and Biological Reaction Engineering, Spring
2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on
[DD Month YYYY].
Cell Growth as a Chemical Reaction
Aerobic growth:
Carbon source + O2 + Nitrogen Source → Biomass + Byproducts + H2O +CO2
CH1.8O0.5N0.2
(for a typical microbial culture)
A yield coefficient can be defined:
YA B =
ΔA
ΔB
A=biomass, byproducts, CO2, heat (any of these)
B=carbon source or oxygen
For glucose, Yx s ≈ 0.6 ± 0.1
For oxygen,
Yx O2 ≈ 1.9 ± 0.7
g biomass
g glucose
g biomass
g O2
10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. K. Dane Wittrup
Lecture 3
Page 6 of 6
Cite as: K. Dane Wittrup, course materials for 10.37 Chemical and Biological Reaction Engineering, Spring
2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on
[DD Month YYYY].