Field guide to functions for biologists
Functions and data can be explored at http://www.shodor.org/interactivate/activities/MultiFunctionDataFly/
Exponential – Formula for representing the change in a quantity over time when the rate
of change depends on the amount. Examples include:
 Radioactive decay – time course of amount of starting material shows exponential
decay
 First-order rate law phenomena where change ~ k*A = rate constant * amount
 Unconstrained growth – in population, economy, …
Decay
At=A0e-kt : Where At is the amount of decaying substance A at time t, A0 is the starting
amount at time = 0, and k is the rate constant governing the decay = the proportion of the
substance that is lost over some time interval
In function flyer : f(x) = 50*exp(-0.15*x)
Growth
At=A0ekt : Where things are the same as in the decay case, except that, with a positive
exponent, substance is being added as time elapses
Rectangular hyperbola – Formula for representing systems where there is a case of
diminishing returns, in other words increasing the input leads to less and less increase in
the output. Examples include:
 Ligand binding – plot of site occupancy as a function of ligand concentration
o Y = [L] / (KD + [L]), where Y is fractional binding site occupancy, [L] is
ligand concetration, and KD is the dissociation constant or concentration at
which half of the binding sites are occupied
 Rate of enzymatic reaction – rate of production as a function of substrate
concentration
o v = vMax * [S] / (KM +[S]), where v is the reaction velocity and vMax is the
maximum velocity (= turnover number times # of binding sites times
enzyme concentration), [S] is substrate concentration, and KM is the
Michaelis constant for quasi-equilibrium substrate binding
 Spring loading – deformation as a function of force applied
 Vehicle velocity – speed as a function of fuel consumption rate, or as a function
of engine power delivered to the wheels
 Capacitor charging – charge on capacitor as a function of applied voltage
o How would the relationship between current through a resistor versus
applied potential differ?
 Filtration – rate of flow through filter as a function of pressure (maybe this isn’t
as limited (this is more of a resistor than a capacitor
Function flyer version of equation
f(x) = 1*x/(50+x)
set x-axis 0 to 100, y-axis 0 to 5
Sigmoidal function –
 Can represent a combination of an exponential unconstrained growth at low
densities followed by an approach toward a limited carrying capacity at high
densities
 Hill equation – occupancy of allosterically coupled binding sites
o f(x) = xn / (Kd + xn) , where x is the ligand concentration, f(x) is the
fractional occupancy of the binding sites, and n is the Hill coefficient, for
which a positive value indicates positive cooperativity (such that binding
at one site increases affinity at the other sites), and negative values
indicate negative cooperativity. The upper limit of the absolute value of
the coefficient is the number of binding sites
 Logistic - f(t) = 1 / (1 + e-t)
 Gompertz - f(t) = aebe^ct : may possess asymmetrical approach to its two
asymptotes, with the lower asymptote approached much more slowly than upper
Linear – Represents situations where there is a direct dependence between parameters in
a system. The scaling factor, or slope, may be positive or negative.
Quadratic – two parameterizations, one that is mathematically easy to derive, the other
that is parametrically easy to understand
 f(x) = ax2 + bx + c
: this is the standard form
2
 f(x) = a(x-h) + k
: this is the vertex form where h and k are the x and y
coordinates of the vertex respectively. This form is derived from the standard
form by a process called completing the square.
Cosine – periodic functions are extremely important for understanding cyclical
phenomena, of which there are many in biology
Gaussian – The normal curve. This is the most famous of the “bump-shaped” functions.
Computational Mathematics Glossary
Differentiation
Integration
Laplace transformation
Series expansion