Tag: imps

Exponential Growth/Decay Models (Summary)

A quick summary (more details here):

The equation that describes exponential growth is:
Exponential Growth: N = N_0 e^{rt}

where:

  • N = number of cells (or concentration of biomass);
  • N0 = the starting number of cells;
  • r = the rate constant, which determines how fast growth occurs; and
  • t = time.

You can set the r value, but that’s a bit abstract so often these models will use the doubling time – the time it takes for the population (the number of cells, or whatever, to double). The doubling time (td) can be calculated from the equation above by:

t_d = \frac{\ln 2}{r}

or if you know the doubling time you can find r using:

r = \frac{\ln 2}{t_d}

Finally, note that the only difference between a growth model and a decay model is the sign on the exponent:

Exponential Decay: N = N_0 e^{-rt}

Decay models have a half-life — the time it takes for half the population to die or change into something else.

Exponential Cell Growth

The video shows 300 seconds of purely exponential growth (uninhibited), captured from the exponential growth VAMP scenario. Like the exponential growth function itself, the video starts off slowly then gets a lot more exciting (for a given value of exciting).

The modeled growth is based on the exponential growth function:

N = N_0 e^{rt} (1)

where:

  • N = number of cells (or concentration of biomass);
  • N0 = the starting number of cells;
  • r = the rate constant, which determines how fast growth occurs; and
  • t = time.

Finding the Rate Constant/Doubling Time (r)

You can enter either the rate constant (r) or the doubling time of the particular organism into the model. Determining the doubling time from the exponential growth equation is a nice exercise for pre-calculus students.

Let’s call the doubling time, td. When the organism doubles from it’s initial concentration the growth equation becomes:

2N_0 = N_0 e^{r t_d}

divide through by N0:

2 = e^{r t_d}

take the natural logs of both sides:

\ln 2 = \ln (e^{r t_d})

bring the exponent down (that’s one of the rules of logarithms);

\ln 2 = r t_d \ln (e)

remember that ln(e) = 1:

\ln 2 = r t_d

and solve for the doubling time:

\frac{\ln 2}{r} = t_d

Decay

A nice follow up would be to solve for the half life given the exponential decay function, which differs from the exponential growth function only by the negative in the exponent:

N = N_0 e^{-rt}

The UCSD math website has more details about Exponential Growth and Decay.

Finding the Growth Rate

A useful calculus assignment would be to determine the growth rate at any point in time, because that’s what the model actually uses to calculate the growth in cells from timestep to timestep.

The growth rate would be the change in the number of cells with time:

\frac{dN}{dt}

starting with the exponential growth equation:

N = N_0 e^{rt}

since we have a natural exponent term, we’ll use the rule for differentiating natural exponents:

\frac{d}{dx}(e^u) = e^u \frac{du}{dx}

So to make this work we’ll have to define:

u = rt

which can be differentiated to give:

\frac{du}{dt} = r

and since N0 is a constant:

N = N_0 e^{u}

\frac{dN}{dt} = N_0 e^{u} \frac{du}{dt}

substituting in for u and du/dt gives:

\frac{dN}{dt} = N_0 e^{rt} (r)

rearranging (to make it look prettier (and clearer)):

\frac{dN}{dt} = N_0 r e^{rt} (2)

Numerical Methods: Euler’s method

With this formula, the model could use linear approximations — like in Euler’s method — to simulate the growth of the biomass.

First we can discretize the differential so that the change in N and the change in time (t$) take on discrete values:
\frac{dN}{dt} = \frac{\Delta N}{\Delta t}

Now the change in N is the difference between the current value Nt and the new value Nt+1:

Now using this in our differentiated equation (Eq. 2) gives:

\frac{N^{t+1}-N^t}{\Delta t} = N_0 r e^{r\Delta t}

Which we can solve for the new biomass (N^t+1):

N^{t+1}-N^t = N_0 r e^{r\Delta t} \Delta t

to get:
N^{t+1} = N_0 r e^{r\Delta t} \Delta t + N^t

This linear approximation, however, does introduce some error.

The approximated exponential growth curve (blue line) deviates from the analytical equation. The deviation compounds itself, getting worse exponentially, as time goes on.
The approximated exponential growth curve (blue line) deviates from the analytical equation. The deviation compounds itself, getting worse exponentially, as time goes on.

Excel file for graphed data: exponential_growth.xls

VAMP

This is the first, basic but useful product of my summer work on the IMPS website, which is centered on the VAMP biochemical model. The VAMP model is, as of this moment, still in it’s alpha stage of development — it’s not terribly user-friendly and is fairly limited in scope — but is improving rapidly.