Straight-line function

The simplest polynomial is the straight line, with equation:

where is the value of when , and is the slope, that is, the change in associated with a one-unit increase in . When , increases as increases; when , it decreases.

Because of their simplicity, straight lines are mainly used to approximate biological responses over a limited range of the predictor. Should you need it, an example of fitting a straight line to research data is given elsewhere at this link.

Quadratic polynomial function

The quadratic polynomial is described by the equation:

This equation describes a U-shaped curve (a parabola), which may open upwards or downwards depending on the sign of . If , the parabola opens upwards; otherwise, it opens downwards.

The parameter represents the value of when , whereas and determine how the response changes as the predictor varies. To better understand the role of these parameters, we can examine the first derivative of Eq. (2). In R, this can be obtained with the D() function, which differentiates an expression with respect to a specified variable:

D(expr = expression(a + b*X + c*X^2), name = "X")

b + c * (2 * X)

The derivative is not constant but varies with the value of and, in particular, represent the variation of around the point . The stationary point, that is, the point where the derivative is zero, occurs at:

This point is a minimum when and a maximum when .

The corresponding response is

In practice, biologists often use only one branch of the parabola to approximate biological responses showing either a concave or a convex trend over a limited range of the predictor. Check this link for an example of fitting.

Exponential function

The most common parameterisation of an exponential function is:

Depending on the sign of , Eq. 3 describes a concave-up monotonically increasing shape (exponential growth; ) or a concave-up monotonically decreasing shape (exponential decay; ).

The parameter represents the value of when , whereas the meaning of can be understood by considering the first derivative:

D(expression(a * exp(k * X)), "X")

a * (exp(k * X) * k)

In algebraic terms, it is:

and thus:

Therefore, represents the relative rate of change (increase/decrease), throughout the domain, which is often known in growth analysis as the Relative Growth Rate (RGR).

Other equivalent parameterisations are:

and:

Equations 3-5 are equivalent, as can be shown by setting and :

Another, slightly different, parameterisation is commonly used in bioassay studies, mainly to describe exponential decay processes:

where corresponds to in the previous models and .

For all the exponential decay equations presented above, as . A lower asymptote, , can be incorporated into Eq. 6 to account for situations in which the response does not approach zero as the predictor tends to infinity:

Exponential functions are often used to describe the growth of populations under non-limiting environmental conditions or the degradation of xenobiotics in the environment (first-order degradation kinetics). In both cases, represents time and is therefore restricted to non-negative values, while (or, equivalently, ) must be different from zero. Check the post at this link for an example of fitting.

Asymptotic function

This function appears in several different parameterisations and is also known as the monomolecular growth function, the Mitscherlich law, or the von Bertalanffy law. Owing to its biological interpretation, the most widely used parameterisation is:

It describes a monotonically increasing, concave-down curve approaching a horizontal asymptote as tends to infinity. The parameter represents the maximum attainable value of (the plateau), while is the value of when (the initial value). The parameter is proportional to the rate at which the response approaches the plateau. Indeed, the first derivative is

D(expression(a - (a - b) * exp (- m * X)), "X")

(a - b) * (exp(-m * X) * m)

Considering Eq.8, we can write:

and then:

The previous expression shows that the relative rate of change in the response (the RGR in growth analysis) is not constant, as it is for the exponential function. Instead, it depends on the attained value of . In particular, the RGR is greatest at the beginning of the process, when is smallest, and gradually approaches zero as approaches the plateau .

Another closely related parameterisation is often encountered, in which is replaced by , by , and by . Some simple algebraic manipulations are also introduced to make the model consistent with the parameterisation commonly adopted for biological assay models:

In some applications, is reparameterised through its logarithm to facilitate model fitting:

For all these equations, setting (or equivalently ) yields a curve passing through the origin. In this case, the most common parameterisation is:

which is known as the negative exponential function.

The asymptotic function is widely used in biology, not only for growth analysis but also, for example, in weed competition studies. The negative exponential function has also been used to model the absorbed Photosynthetically Active Radiation () as a function of leaf area index (). In this context, represents the incident PAR (), while is the light extinction coefficient. Check the post at this link for an example of fitting.

Power function

The power function is also known as Freundlich function or allometric function. The most common parameterisation is:

Its shape is highly flexible and is determined by the value of the parameter (see Figure 1). When , the response increases with and the curve is concave down. When , decreases as increases and the curve is concave up. Finally, when , increases with and the curve is concave up. These three cases are illustrated in Figure 1 using different colours. The function is defined only for and has no horizontal asymptote as .

The biological interpretation of the parameters and is not straightforward. Both influence the slope of the curve, as can be seen by examining the first derivative:

D(expression(a * X^b), "X")

a * (X^(b - 1) * b)

The power function is mathematically equivalent to an exponential function of , since:

The power function (named as the Freundlich equation) is widely used in agricultural chemistry, for example to model the sorption of xenobiotics in soil. It is also used to describe the relationship between the number of plant species and the sampled area (the species–area relationship). Check the post at this link for an example of fitting.

Logarithmic function

This function is linear in :

Because of the logarithmic transformation, the function is only defined for . The parameter determines the shape of the curve. When , the response increases with and the curve is concave down. Conversely, when , decreases as increases and the curve is concave up, as shown in Figure 1.

The interpretation of the parameters is fairly straightforward. The parameter is the value of the response when . Changing while keeping constant simply shifts the curve vertically without altering its shape, so that the difference between any two curves remains constant for every value of (Fig. 2).

The parameter controls the slope of the curve and, more specifically, it is equal to the slope at , as can be seen from the expression for the first derivative:

D(expression(a + b*log(X)), "X")

b * (1/X)

Changing while keeping constant produces curves that all intersect at (Fig. 2).

In biology, logarithmic functions are used, for example, to describe species–area relationships in ecology, enzyme kinetics in biochemistry, and sensory perception in neurobiology. Check the post at this link for an example of fitting.

Rectangular hyperbola

The rectangular hyperbola is commonly known as the Michaelis-Menten function and is usually parameterised as

The curve is monotonically increasing and concave down, approaching the horizontal asymptote (plateau) . It passes through the origin of the axes ( when ). The parameter is the value of that produces a response equal to . Indeed,

which is readily solved to obtain .

The first derivative is:

D(expression( (a*X) / (b + X) ), "X")

a/(b + X) - (a * X)/(b + X)^2

From this expression, we can see that the initial slope (at ) is .

Equation 14 is not defined for and has no biological meaning for . In most biological applications, however, both and are positive.

An equivalent parameterisation is:

In this form, the parameters and are often replaced by and , respectively. Although the response is not defined for , it tends to zero as .

Another common parameterisation includes the initial slope as an explicit parameter because of its biological relevance. This is obtained by dividing both the numerator and denominator by and noting that , so that :

The Michaelis-Menten function is widely used in pesticide chemistry, enzyme kinetics (Eq. 14), biological assay models (Eq. 15), and weed competition studies, where it is used to describe crop yield loss as a function of weed density (Eq. 16). An example of fitting is shown in another post, at this link.

Logistic function

The logistic curve is derived from the cumulative logistic distribution function. It is symmetric about its inflection point and can be parameterised as:

where is the upper asymptote, is the lower asymptote, is the value of at the inflection point, and is proportional to the slope at the inflection point. Because the curve is symmetric, also represents the value of that produces a response halfway between and (commonly referred to as the ED50 in biological assays). The parameter may be either positive or negative and, consequently, the response may either increase or decrease as increases.

This equation is known as the four-parameter logistic model. If appropriate, constraints can be imposed on the parameter values. For example, may be fixed at 0, yielding the three-parameter logistic model. If, in addition, is fixed at 1, the model reduces to the two-parameter logistic model.

Gompertz function

The Gompertz curve can be parameterised in many different ways. I prefer a parameterisation that closely resembles that of the logistic function:

Unlike the logistic function, the Gompertz curve is not symmetric about its inflection point. It exhibits a longer lag phase at the beginning, followed by a progressively steeper increase before gradually approaching the upper asymptote. The parameters have essentially the same interpretation as those of the logistic function, except that , the abscissa of the inflection point, does not correspond to the value of producing a response halfway between and .

As with the logistic function, four-, three-, and two-parameter Gompertz models can be obtained by constraining one or both asymptotes.

Modified Gompertz function

We have seen that the logistic curve is symmetric about its inflection point, whereas the Gompertz curve exhibits a longer lag phase at the beginning, followed by a progressively steeper increase. A different asymmetric pattern can be obtained by modifying the Gompertz function as follows:

The resulting curve increases rapidly at the beginning but gradually slows down as it approaches the upper asymptote. As with the logistic and Gompertz functions, one or both asymptotes can be constrained ( and/or ), giving rise to four-, three-, and two-parameter versions of the modified Gompertz model.

Log-logistic function

The log-logistic curve is symmetric with respect to . A log-normal curve has a very similar shape, although it is used much less frequently. In biological assays (and also in germination studies), the log-logistic function is commonly parameterised as

The parameters have the same interpretation as in the logistic function. In particular, represents the value of that produces a response halfway between and (the ED50). It is easy to show that the above equation is equivalent to

Like the logistic function, the log-logistic model can be fitted in four-, three-, or two-parameter versions by constraining one or both asymptotes. It is widely used to describe dose-response relationships in biological assays, seed germination, and crop growth.

Weibull function (type 1)

The Type I Weibull function is the logarithmic counterpart of the Gompertz function, being defined on rather than on . It is parameterised as

The parameters have essentially the same interpretation as those of the other sigmoidal functions presented above. In particular, and are the lower and upper asymptotes, respectively, while is the value of corresponding to the inflection point. Unlike the log-logistic function, however, does not correspond to the ED50.

Weibull function (type 2)

The Type II Weibull function is closely related to the Type I Weibull function, but it describes a different type of asymmetry, analogous to that of the modified Gompertz function:

The parameters have the same interpretation as those of the Type I Weibull function. In particular, and are the lower and upper asymptotes, respectively, while is the value of at the inflection point. As with the Type I Weibull function, does not correspond to the ED50. One or both asymptotes may be constrained, giving rise to four-, three-, and two-parameter versions of the model.

Another ‘flexible’ sigmoid

I would also like to mention another sigmoidal function that has been widely used in biology because of its supposed flexibility: the Richards function:

In this model, the degree of asymmetry is controlled by the parameter . When , the function reduces to the logistic model, whereas, as decreases below 1, the curve progressively resembles a Gompertz function. Figure 4 illustrates the different shapes obtained for different values of .

Despite this apparent flexibility, the Richards function has poor statistical properties for parameter estimation. In particular, the additional parameter often causes strong correlations among the estimates, making the fitting process unstable and the parameter estimates difficult to interpret. Ratkowsky (1990) described it as having “more undesirable nonlinear regression behaviour than almost any nonlinear regression model in common use.” For these reasons, the Richards function will not be considered further in this blog.

Peaked sigmoidal function

The log-logistic decreasing curve can be modified to account for possible hormetic effects at low doses, by combining mathematical switching functions (Schabenberger and Pierce, 2002; pag. 275). The most widespread parameterisation, that was originally devised by Brain and Cousens (Brain, P., Cousens, R., 1989. An equation to describe dose responses where there is stimulation of growth at low doses. Weed Research 29, 93–96) in a slightly different form, is:

The parameters have the same interpretation as in the log-logistic curves, but does not represent the response half-way between the lower and higher asymptote and represents the size of the hormetic effect, which increases as increases, while corresponds to the situation of no hormesis (and the function reduces to a log-logistic.

Bragg function

This function is connected to the normal (Gaussian) distribution and has a symmetric shape with a maximum equal to , that is reached when and two inflection points. In this model, relates to the slope at the inflection points; the response approaches 0 when approaches :

If we would like to have lower asymptotes different from 0, we should add the parameter , as follows:

The two Bragg functions have proven useful in applications relating to the science of carbon materials.

Lorentz function

The Lorentz function is similar to the Bragg function, although it has worse statistical properties (Ratkowsky, 1990). The equation is:

We can also allow for lower asymptotes different from 0, by adding a further parameter:

Beta function

The beta function derives from the beta density function and it has been adapted to describe phenomena taking place only within a minimum and a maximum threshold value (threshold model). One typical example is seed germination, where the germination rate (GR, i.e. the inverse of germination time) is 0 below the base temperature level and above the cutoff temperature level. Between these two extremes, the GR increases with temperature up to a maximum level, that is reached at the optimal temperature level.

The equation is:

where is the maximum level for the expected response , and are, respectively, the minumum and maximum threshold levels, is the abscissa at the maximum expected response level and is a shape parameter. The above function is only defined for and it returns 0 elsewhere.