The responding entities (molecules, cells, seeds, organisms, etc.) have thresholds for sensing factors (X) to which they can respond. For a factor promoter (+) of a response, such as a hormone, there will be a minimum threshold concentration that will trigger a response, whether for a cell or for a seed. This threshold sensitivity of factor X is termed the base level for that factor, or X_b. If the factor is an inhibitor (–) of the response, then the factor level that just prevents the response from occurring is termed the base level.

The rate at which the response is activated or progresses is proportional to the difference between the factor level (X) and the base threshold sensitivity to it ( X_b ):

(+) ( X – X_b ) or (–) ( X_b – X ) .

The product of this difference in factor level in excess or below the base threshold (X – X_b or X_b – X , respectively) and the time to response (t) is a constant, called the time constant (θ_X ), resulting in the following equation for the model:

(+) θ_X = ( X – X_b ) t or (–) θ_X = ( X_b – X )t .

Thus, as the factor level difference increases, the time to occurrence of the regulated process decreases proportionately to keep θ_X constant. Since rates are the inverse of the time required for something to happen, this can also be shown as:

(+) Response rate = 1 / t = ( X – X_b ) / θ_X or

(–) Response rate = 1 / t = ( X_b – X ) / θ_X .

The final feature of PBT models is that the values of the base thresholds vary among the individual responding entities. That is, X_b varies among individuals, generally in a normal distribution that can be defined by its mean and standard deviation. This results in the final equation for the PBT model, where X_b (i) indicates the normal distribution of X_b values across individuals (or fraction) i, and t_i is the time to response of individual i :

(+) Response rate = 1 / t_i = [ X – X_b(i) ] / θ_X or

(–) Response rate = 1 / t_i = [ X_b(i) – X ] / θ_X .