PBTModels: The rolling ball analogy
Population-based threshold (PBT) models are conceptually simple, but it can be hard to visualize how the different components fit together and influence the predicted behaviors. We have developed different analogous examples that can help users understand how the different components interact. Here, we illustrate the Rolling Ball Analogy in a series of animated videos.
The PBT models have only a few basic components:
Factors and base levels
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 Xb. 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.
Response rate
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 ( Xb ):
(+) ( X – Xb ) or (–) ( Xb – X ) .
Time constant
The product of this difference in factor level in excess or below the base threshold (X – Xb or Xb – X , respectively) and the time to response (t) is a constant, called the time constant (θX ), resulting in the following equation for the model:
(+) θX = ( X – Xb ) t or (–) θX = ( Xb – X )t .
Thus, as the factor level difference increases, the time to occurrence of the regulated process decreases proportionately to keep θX 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 – Xb ) / θX or
(–) Response rate = 1 / t = ( Xb – X ) / θX .
Base thresholds variation
The final feature of PBT models is that the values of the base thresholds vary among the individual responding entities. That is, Xb 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 Xb (i) indicates the normal distribution of Xb values across individuals (or fraction) i, and ti is the time to response of individual i :
(+) Response rate = 1 / ti = [ X – Xb(i) ] / θX or
(–) Response rate = 1 / ti = [ Xb(i) – X ] / θX .
With this brief background, the videos above illustrate how each component of this model works to influence the response fractions and rates in relation to variation in factor levels and time constants. For examples and information on applications of PBT models to seed germination behavior and other biological processes, see the references below.
Bradford KJ, Bello P (2022) Applying population-based threshold models to quantify and improve seed quality attributes. In J Buitink, O Leprince, eds, Advances in Seed Science and Technology for More Sustainable Crop Production. Burleigh Dodds Science Publishing, Cambridge, UK
Bradford KJ (2018) Interpreting biological variation: seeds, populations and sensitivity thresholds. Seed Sci Res 28:158-167. Doi:10.1017/S0960258518000156