Choosing a Mathematical Form for Enzyme Maturation

Why the Mathematical Form Matters

A PBPK model may need to represent how an enzyme’s activity changes from birth toward an adult level. The mathematical function used for this change affects predictions at ages between observed or specified developmental stages. A curve that rises gradually from birth, one that rises sharply after a delay, and one that changes in straight-line segments can all reach the same adult value while implying different activity at a particular pediatric age.

Choosing a function is therefore a choice about the shape and assumptions of the developmental trajectory, not merely a choice of equation. The earlier material on gestational, postnatal, and postmenstrual age establishes why the age clock must be specified. In the equations below, \(a\) denotes the chosen age in years since birth, for illustration. An application using another age coordinate must define its origin and units consistently.

These forms commonly assume a single, nondecreasing trajectory from birth to adulthood, a fixed choice of age scale, and a meaningful adult reference. They do not automatically describe every enzyme. Activity could be nonmonotone, differ across developmental subgroups, or be higher before birth than in adults. In such cases, a simple monotone rise from birth is not an adequate representation. The maturation multiplier also describes an age-related change in relative enzyme activity; it should not be mistaken for a complete account of every age-related change in a PBPK model.

Sigmoid Emax and Hill Forms

A common smooth formulation is the sigmoid Emax function. It starts at \(M_0\), rises toward \(1\), and allows the transition to be more or less steep. For \(a\geq 0\), one normalized form is:

Here, \(T_{50}>0\) is an age, in the same units as \(a\), and \(\gamma>0\) is a dimensionless shape parameter. At \(a=0\), the fraction is zero, so \(M(0)=M_0\). At \(a=T_{50}\), the numerator and the age-dependent part of the denominator are equal, so the fraction is \(1/2\). Thus \(M(T_{50})\) is halfway between the birth value and the adult value, not necessarily equal to 50% of adult activity:

The exponent \(\gamma\) controls the curve’s shape. With \(\gamma=1\), the rise has the ordinary Emax or rectangular-hyperbola form. Larger values produce a more pronounced transition around \(T_{50}\); smaller positive values produce a less sharply localized transition. The function remains bounded between \(M_0\) and \(1\) for these parameter restrictions.

The Hill form is often written using a maximum, baseline, and half-saturation age. With \(M_{\max}=1\) and Hill coefficient \(h\), it is:

When \(M_{\max}=1\), \(K=T_{50}\), and \(h=\gamma\), this is the same equation as the sigmoid Emax form above. In many applications, “sigmoid Emax” and “Hill” name the same mathematical family, with differences in parameter notation or in whether the maximum is explicitly estimated. The labels alone do not establish that the models differ; compare the equations and parameter definitions.

Exponential Maturation

An exponential form represents a smooth rise that slows as it approaches the adult value:

The rate constant \(k\) has units reciprocal to the age unit; if age is in years, \(k\) is in years\(^{-1}\). The function equals \(M_0\) at age zero and approaches 1 as age increases. It does not attain 1 at a finite age. Its shape has no separately adjustable midpoint or steepness parameter: \(k\) controls the pace of the rise.

Piecewise Maturation

A piecewise function specifies activity at selected ages and defines the trajectory between them. A common choice is linear interpolation between ordered age-activity pairs \((a_j,M_j)\). For ages \(a_j\leq a\leq a_{j+1}\), the interpolated value is:

If the specified values increase from \(M_0\) to 1, this construction gives a nondecreasing curve between the knots. It is continuous at each knot, but its slope can change abruptly there. A final segment may be defined as a plateau at 1 after a selected age. Unlike the smooth asymptotic curves, that version reaches the adult value at a finite age by construction.

Comparing the Forms and Checking Their Use

The forms differ in flexibility, interpretation, and behavior between developmental ages. A smooth sigmoid or exponential function supplies a continuous trajectory across the age range. A piecewise function makes selected age-activity anchors transparent, but interpolation choices determine the values between them. The most convenient function is not automatically the most biologically appropriate one.

Several checks help reveal a poor choice or implementation. Confirm that the age variable has the intended origin and units, that \(M(0)\) equals the specified birth value, and that the predicted values remain within biologically intended bounds. Check whether the function is monotone when monotonicity is assumed, and whether it reaches or only approaches the adult reference as intended. Plotting the curve over the full age range is a simple way to detect an unintended early jump, implausible plateau, discontinuity, or unit error.

A further check is to distinguish a true shape difference from a naming difference. A Hill equation with a baseline and adult maximum can be algebraically identical to sigmoid Emax. Conversely, the exponential form is not generally equivalent to either: it has a different shape and different parameter interpretation. Piecewise interpolation is also not simply a smooth approximation unless its knot spacing and slopes make it so.

Choosing a Form Without Over-Interpreting It

Use the simplest function that represents the intended developmental pattern over the age range relevant to the PBPK application. Sigmoid Emax or Hill forms are useful when a transition with an interpretable midpoint and adjustable steepness is needed. The exponential form is compact when a gradual approach with a single rate parameter is a reasonable description. Piecewise functions can be useful when specific age anchors are central to the representation, provided the behavior between anchors is made explicit.

Each form has limitations. A smooth curve can suggest more certainty about intermediate ages than the information supporting it warrants. A piecewise curve can create artificial slope changes at knots. A monotone curve may be unsuitable if activity rises and later declines. A model with a fixed adult asymptote of 1 is also inappropriate if the chosen adult reference does not represent the relevant mature level. Mathematical convenience does not establish biological truth; the predicted trajectory should be examined against the intended biology and evidence.