A phasic model for the analysis of sigmoid patterns of growth

J.H. Lieth and P.R. Fisher

Department of Environmental Horticulture, University of California, Davis, CA 95616-8587


R.D. Heins

Department of Horticulture, Michigan State University, East Lansing MI 48824-1345

Many organisms go through various distinct phases of growth. Quantification of such patterns is achieved by tracking some measurable size or weight variable over time. The most common pattern seen under relatively stable conditions is a "sigmoid" pattern where this variable sequentially goes through phases that appear to be exponential, then linear, and finally asymptotic to some upper limit.

The Richards function (RF) is a commonly-used model for such patterns which represents sigmoid patterns very well but deals with the three growth phases only approximately.

In this project we: 1. developed a model for sigmoid growth which uses a phasic approach to represent growth patterns explicitly, 2. compared this model against the Richards function and 3. applied this model to the stem elongation of poinsettia.

Model Derivation

If the desired growth pattern of the state variable y goes through three distinct phases: an exponential phase, a linear phase, and a saturation phase, then this can be expressed mathematically by

Eqn 1

where t0 is the starting time and t1 and t2 are the transition points.

This figure shows how the three phases of growth are represented mathematically by the model.
Fig 1
It is desirable to have the pieces join at the transition points (i.e. continuity for y) and for these transitions to be smooth (continuity of the first derivatives at t1 and t2). Thus we have the restriction:

eqn 2

where f(t) represents the first derivative of F(t). This means that b0 = Fexp(t1), b2 = Flin(t2), b1 = fexp(t1), b3 = fexp(t1)/gamma, so that:

eqn 3

This can be written as

Eqn 4

By simplifying this, and reparameterizing so that the difference t2-t1 is represented as the parameter epsilon, we get the final form with the 5 parameters: alpha, beta, gamma, t1 and epsilon:

Eqn 5

Fig 2 This figure shows how each of the parameters relate to the shape of the curve.

The five parameters in this model have the following meaning: alpha is the initial size of y, beta represents the intrinsic growth rate of the exponential phase, t1 is the transition point between the first two phases, epsilon is the length of the linear phase, and gamma is the intrinsic saturation rate.

Comparison with the Richards Function

The RF differential equation is

Eqn 6

and has the solution:

Eqn 7

where y0 is the initial value of y (i.e. R(t0)), yf represent asymptotic value of R (as t gets infinitely large), and k and n lack biological meaning but are related to the shape of the curve and its inflection point.

The phasic model was compared with R(t) for a wide range of possible shapes of the RF. To do this, the scale of the RF was fixed and only the shape was allowed to vary by constraining the analysis to those cases where y0= 10, yf= 100, t=0 coinciding with y0.

We also wanted R(100)=0.996 yf, to assure that a substantial number of points near the upper asymptote enter into the analysis. This means that for any shape selected with the parameter n we have k fixed at

Eqn 8

For this analysis, n was varied from 0.1 to 5 in increments of 0.1. For each value of n, a sequence of 200 coordinates, spaced equally along the t-axis from t=0 to t=100, were computed with the RF in a SAS data step (SAS Institute, 1988). Then PROC NLIN of SAS was used to numerically fit F(t) to these data.

Virtually any trajectory traversed by the RF can be mimicked closely by F(t). The deviations at any point are less than 4% so that the trajectories are very close together (see Figure below). Even in the worst fits (i.e. for n =5) this amounts to fitting 99.93% of the variation in the data generated from the RF.

Fig 3

Subtraction of F(t) from the RF data (dashed lines) shows that the minute biases vary in location along the pattern with increasing n. Clearly the curves are very close together, indicating that F(t) can always replace the RF due to its flexibility to fit a wide range of shapes.

The analysis of this model showed it to be applicable wherever the Richards function could be used while providing improved flexibility over the Richards function.

Fig 4 The residual sum of squares of the fitting decreased as n increased.

The fitted parameters alpha, beta, gamma, t1 and epsilon also showed patterns with n.

In its most general form the function developed here has 5 parameters compared with 4 for the RF. This increased complexity buys a substantial improvement in control over the shape of the curve. In addition to accommodating all situations where the RF is suitable, the model is suitable for a wide range of additional possibilities including exponential, linear, or saturation patterns, as well as combinations of any two of these.

Example: Poinsettia Stem Elongation

For stem elongation of poinsettia (Euphorbia pulcherrima Klotz.) an extended linear phase and gradual slope immediately after planting had been observed. Stem elongation data were obtained and modeled with the phasic model and the RF.

Fig 5

The phasic model tracked the data more accurately than the Richards function particularly between day 10 and 30.


The phasic growth model developed here provides a new useful tool for the analysis of growth. Thee analysis showed the model to be applicable wherever the Richards function could be used. It was able to improve on the fit, particularly in a case (poinsettia) where there was a lengthy lag phase followed by a long linear phase.

In addition to having parameters with biological meaning, the model is also more suitable for use in simulations where conditions are not constant. This is of particular importance when implementing such models in greenhouse environment decision support systems.

For more information or feedback contact jhlieth@ucdavis.edu