Contribution to the celebration of 100 years of chemical engineering at RPI

Cellular Growth RatesHenry R. Bungay 3rd, Emeritus Professor

OverviewGrowth 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.

Be precise:µ is the specific growth rate coefficient

IntroductionNatural 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 bioprocessingHolding a bioprocess at a product formation phase instead of a growth phaseFavoring desired cells over competing cellsMathematical modeling to provide clues to process improvementAccolades for unraveling fundamental biochemistry

Growth MathematicsThe 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.

Background conceptsWhile 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 phasesClick here to review endogenous metabolism

Monod EquationKs 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.

Intermediate Ks Low Ks Large Ks

Here are some graphs drawn using the Monod equation:

Alternatives to MonodLet’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.

Don’t use the Monod equation

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.

Mass BlancesChemical engineers dwell on mass balances. The fundamental equation is:rate of change = input – output ± reactionfor cell mass designated x g/L,dx/dt V = Fi xi – Fo xo + µ x Vfor growth-limiting nutrient S g/L,dS/dt V = Fi Si – Fo So - µ x V / Ywhere V = vessel volume, LFi & Fo are the input and output flows, L/mxi, xo, Si, So are the input and output concentrations, g/L. Y is the proportionality.

Yield Coefficient, YQuick 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.

Simplification for no flows

rate of change = input – output ± reaction

dx/dt V = µ x V dx/dt = µ x

dS/dt V = - µ x V / Y 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

Consider two nutrientsAll 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.

Either-OrWhile 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.

Plots for either - or

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.

Double Monod Model

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

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?).

Reduction to absurditySuppose 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 PO4

Ξ, 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.

Model Using Weighting FactorsOur 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

fitsreal data quite well.

Substrate Inhibition

Andrews equation,Ki = inhibition coefficientToo 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.

That’s All

Use the esc (escape) key to exit the slide show.

The rest of the slides are background materials.

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.

Phases of microbial growth

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.

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Growth Limiting NutrientThe 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".

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Maintenance Energy

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