Contribution to the celebration of 100 years
of chemical engineering at RPI
Cellular Growth Rates
Henry R. Bungay 3rd, Emeritus Professor
Be precise:
µ is the
specific
growth rate
coefficient
Growth rate is fundamental to all
bioprocessing with living cells. A key
concept is growth-limiting nutrient, the
one in lowest proportion to the others. It
will be exhausted first and is the focus for
growth rate control. When limitation can
shift from one nutrient to another, the
equations for the specific growth rate
coefficient become more complicated. If
one nutrient is inhibitory to growth as its
concentration rises, additional terms are
needed in these equations.
Natural nutrients and nutrients for commercial production are seldom
pure. For laboratory research, the nutrients may be well-defined.
Compounds or mixtures of compounds contain carbon, hydrogen,
nitrogen, and other elements in various proportions and may supply
precursors at several locations in metabolic pathways. A rigorous
treatment of growth rate control should consider the exact biochemical
compositions of all nutrients, but this is impractical. We simplify the
complicated patterns of nutrition by speaking of the "carbonaceous
ingredient", the "nitrogeneous ingredient", and the like.
Better understanding of what determines growth rate can reap rich rewards by:
Discovering how to speed up bioprocessing
Holding a bioprocess at a product formation phase instead of a growth phase
Favoring desired cells over competing cells
Mathematical modeling to provide clues to process improvement
Accolades for unraveling fundamental biochemistry
The Monod equation (rhymes with throw)
This is a great starting point for modeling growth but has fundamental
weaknesses that will be discussed later. It comes from considerations of
enzymatic catalysis. The analogy is that an enzymatic reaction can go no faster
when all of the active sites on an enzyme are saturated, and cells can grow no
faster when the nutrients are in great excess.
While mammalian cells and plant cells are the hottest topics at
present, much of the research on growth rates has employed
microorganisms because they are relatively easy to culture and
grow rapidly while cell cultures are very easily contaminated and
grow relatively slowly on expensive culture media. The following
links should be skipped if you already know a lot about microbial
growth:
Click here to review concepts about growth phases
Click here to review endogenous metabolism
Ks is called the half-saturation coefficient because it corresponds to the
concentration at which µ is one-half of its maximum. This can be seen from
the Monod equation by setting S equal to Ks.
Here are some graphs drawn using the Monod equation:
Intermediate Ks
Low Ks
Large Ks
Let’s ignore the several simple equations that relate growth rate to
nutrient concentration because they are curve-fitting exercises.
The most interesting alternatives to the Monod equation are
modifications that consider shifting from one growth-limiting nutrient to
another.
One glaring weakness of the Monod equation is that its graph passes
through zero, and we know that there can be no growth until endogenous
metabolism has been satisfied. This is not important for many situations
in the laboratory or in commercial processes because the nutrient
concentrations remain relatively high.
However, our natural environment consists of countless situations where
nutrients are scarce, and cells compete to survive.
Obviously it is worthless in the death phase of a process
when cells are not growing.
Models based on it may be useful during the logarithmic
growth phase, but its coefficients can change markedly as
cells age. Those who conduct research and development in
industry report that their models of the bioprocesses
seldom rely solely on the Monod equation.
Chemical engineers dwell on mass balances. The
fundamental equation is:
rate of change = input – output ± reaction
for cell mass designated x g/L,
dx/dt V = Fi xi – Fo xo + µ x V
for growth-limiting nutrient S g/L,
dS/dt V = Fi Si – Fo So - µ x V / Y
where V = vessel volume, L
Fi & Fo are the input and output flows, L/m
xi, xo, Si, So are the input and output
concentrations, g/L. Y is the proportionality.
Quick digression about Y, the yield coefficient.
It is the ratio of substrate consumed to cell mass created.
This seems logical, but it assumes that the symbol X that
we use for cell mass represents all cells. It does not
because young cells differ from old cells. Even more
complicated is the behavior of starving cells and cells that
are thriving in rich medium. Many types of cells in rich
medium try to store nutrients as rapidly as possible for
later use, and this results in more cell mass per unit of
substrate consumed than for cells that use it all to stay
alive in dilute media.
rate of change = input – output ± reaction
dx/dt V = µ x V
dS/dt V = - µ x V / Y
dx/dt = µ x
dS/dt = - µ x /Y
Cell mass increases while nutrient is consumed with Y as
the yield coefficient, the amount of S to make x
Because µ depends on the concentration S (Monod
equation), the rates of change fall to zero as S is used up
All nutrients will be used for cell growth. Let’s just consider two.
dx/dt = µ x
dS/dt = - µ x /Ys
dN/dt = - µ x /Yn
For example, S stands for concentration of sugar and N stands for the
concentration of the nutrients containing nitrogenous components. Note
that the yield coefficients Ys and Ys are different. For the carbohydrate
nutrition, the value will be roughly six times that for nitrogen because cells
have much more organic carbon than organic nitrogen.
We could add a mass balance equation of every element that is part of a
cell, but carbon and nitrogen are most relevant.
Having S and N in exactly the correct proportions is extremely rare, and
one will be the growth-limiting nutrient and be exhausted first.
While not common in an industrial bioprocess, there
are systems in which growth rate can be limited by
more than one nutrient. For example, a waste
treatment unit could have feed rich in carbohydrates
and low in nitrogen and later be faced with feed low in
carbohydrates and rich in nitrogen. The nutrient in
lowest proportion would change, and there may be
transitions during which more than one is below a
concentration that would give no rate limitation. We
can also envision a process operated intentionally at
low concentration of nitrogenous nutrients to restrict
growth and at low sugar concentration to favor a
particular pathway.
The blue and green lines are for each potentially limiting nutrient
at the concentration shown on the abscissa. Excess means that
the other nutrient is at a concentration that is not limiting.
Instead we have used the slider bars to select a percentage of the
reference concentration for each nutrient.
This approach is very common for models of microbial
systems subject to growth rate dependent on two
nutrients but is fundamentally flawed. S is the
concentration of one main nutrient (sugar?) and N is
another (nitrogenous nutrient?).
Bader, F.G.,(1978) "Analysis of Double Substrate Limited Growth",
Biotechnol. Bioeng. 20: 183-202
Bader, F.G., (1982) "Kinetics of Double Substrate Limited Growth" in
Microbial Population Dynamics, ed. M.J. Bazin, CRC Press, 1-32
Suppose that we have a medium that has
several potentially growth-limiting factors
such as P, S, N, C, Mg, etc. They may be
needed as PO4Ξ, NH4+, and the like. If each
were at 80 % of the concentration that
would give the maximum growth rate,
extension of the Double Monod model
would predict that the specific growth
rate coefficient would be 0.8 multiplied by
itself as many times as there were
limiting nutrients. This gives a ridiculous
result.
Our group searched for a better model but wanted to use exactly the
same coefficients as the double Monod model. It is not at all difficult to
program on a computer. The concept is that growth rate depends on how
far you are from the half-saturation concentration. Growth rates under
dual substrate limitation may be expressed by weighting the contributions
of individual nutrient limitations:
This approach fits
real data quite well.
Andrews equation,
Ki = inhibition
coefficient
Too much of anything
tends to be bad, and
some nutrients
essential for growth
are toxic at higher
concentrations. This
equation is a good
first start for
modeling.
There is much more for true understanding of cellular
growth rates. For example, the research on what cells
sense, the biochemistry of how they respond, and the
rates of interdependent enzymatic reactions.
Research on the genes involved is in its infancy.
This presentation provides an introduction and may
whet your appetite to learn more.
Use the esc (escape) key to exit the slide show.
The rest of the slides are background materials.
Go Back
Growth begins after a lag phase during which the cells acclimate. They double,
double, and double over and over during the logarithmic phase. When the
limiting nutrient is exhausted, nothing much happens in the stationary phase
before they begin to die.
The growth-limiting nutrient in low proportion to the others will be
exhausted first. There are many ingredients that must be present in living
cells. Some of these are: C - carbon, S -sulfur, N - nitrogen, etc. It is more
convenient to deal with compounds that supply these elements. Commonly,
we refer to the carbohydrate component such as glucose as the growth
limitation for carbon although some carbon may be derived from a
compound that contains carbon in addition to another element such as
nitrogen. With nutrients that are defined, it is fairly easily to deal with one in
low proportion to the others as the growth-limiting nutrient. With complex
nutrients such as soybean meal or distillers solubles, it is not
straightforward to decide where the carbon and nitrogen are coming from
and to state the growth-limiting nutrient.
One way to identify what is growth limiting is to reduce concentrations of
the various nutrients systematically one at a time. When this produces little
or no effect, the nutrient that is being tested is probably in excess. If there is
a definite effect, the nutrient is very likely the "growth-limiting nutrient".
Back
Organisms consume nutrients just to remain
alive.
This is called endogenous metabolism.
Individual cells do not multiply until there is
energy greater than that for not dying.
Go Back