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While it is a daunting challenge in current biology to understand how the underlying network of genes regulates complex dynamic traits, functional mapping, a tool for mapping quantitative trait loci (QTLs) and single nucleotide polymorphisms (SNPs), has been applied in a variety of cases to tackle this challenge. Though useful and powerful, functional mapping performs well only when one or more model parameters are clearly responsible for the developmental trajectory, typically being a logistic curve. Moreover, it does not work when the curves are more complex than that, especially when they are not monotonic. To overcome this inadaptability, we therefore propose a mathematical-biological concept and measurement,

Whether there are different genes responsible for the formation of a trait and how these genes regulate the trait are of fundamental importance biologically, agriculturally, and/or medically. Quantitative traits, or characteristics varying in degree, can be attributed to the effects of genes and their environment [

It should be noted, however, that the single-valued traits are only a portion of the numerous traits, of which many others change with time or other independent variables and are the so-called complex traits. In fact, measurement values in the mature state provide much less information than the growth process leading to it [

Ma et al. [

We now briefly review how functional mapping differentiates developmental trajectories. First of all, the growth process of an individual or an organ can be described by a growth curve, a function of a measurable variable against time. Theoretically, a growth curve may provide infinite amount of information, unlike a single measurement value in a mature state. For example, we consider two bunches of growth curves which are described by the following two equations and illustrated in Figure

Growth curves with different parameter values. (a) is a bunch of curves described by (

In practice, discrete values of the developmental process are measured and collected, based on which functional mapping recovers the process by describing it with a curve which is determined by one or more parameters (

If all growth curves can be described by a function with one or more varying parameters, then we can employ these parameters to be the characteristic values, which is the essence of functional mapping. Unfortunately, no function is qualified for describing all growth types. Specifically, Figure

Four major types of growth curves of the organs and the body as a whole, from birth to 20 years [

The diversity of growth curves gives rise to a problem: how can we differentiate them with one or more characteristic values? An important characteristic value,

As is shown in Figure

It is common sense that a growth curve is continuous and smooth, but in order to elucidate the concept of

An imaginary scenario about growth. The time interval of the growth is from 0 to 9, and the measurement values are all 2 at the beginning point and are all 7 at the end point.

Take in Figure

Compared with (c), the individual indicated by (d) grows “later” since, at the “earlier” time point 2, it increases less, while, at the “later” time point 6, it increases more. Intuitively, the earliness degree of the individual indicated by (c) is more than that of the individual by (c). But we need to quantitatively measure the earliness degree to systematically reflect the difference and comparison. Obviously, two factors are to be considered: increased height and the time span from the time point when increasement occurs to the end time point, and therefore we use their product to represent the earliness degree.

On the basis of the analysis and discussion above, we are now able to calculate the earliness degree of individual (c) as the sum of the areas of two rectangles, one being 4 (increased height) by 7 (time span to the end) and the other being 1 (increased height) by 3 (time span). The area sum is

Similarly, the earliness degree of (d) is calculated as

Using the same method we can calculate the earliness degree of the other 4 individuals, with those of (a) and (b) being trivially 1 and 0, respectively. But the cases of (e) and (f) are more complicated, since the height of (e), before reaching the end time point, has increased to a value 8, a greater value than that of the mature state, 7; and the height of (f) has decreased to a value 1, a smaller value than the beginning value 2. Nevertheless, the earliness degree of (e) can be calculated as

We denote the quantitative earliness degree as

To give the definition, we do not require a growth curve to be globally differentiable, but it is currently required to be piecewise differentiable, which as we will see later is not necessarily met. And we hence give the definition of

Suppose that the growth curve is a continuous function

It should be noted that

How early or how late the growth rate

From the definition above, we can derive several of

If the growth curve function

Proposition

The conclusion of Proposition

We can illustrate the proof with Figure

Illustrations for the proposition proofs. (a) is a growth curve strictly monotonically increasing. (b) is a nonmonotonic growth curve. (c) is a piecewise smooth growth curve. (d) is a resulting growth curve by smoothening that in (c) near the unsmooth internal point.

The conclusion of Proposition

Propositions

Typically, the measurement value in the growth process is between the value at the beginning and that at the end. And we have still another proposition for this situation.

If the growth curve function

But

Proposition

The function parameter

Simulations validating the effectiveness of

For each of the 25 values of

The relation between the values of the parameter

In some cases, we may use

Applying (

This example illustrated that

In order to differentiate growth curves by function parameters, we have to assume the function type first and then calculate parameters making the function fit the successively collected measurement values best. The resulting parameters do not work well if the function does not fit the data well.

In fact, spline interpolation performs well to find a smooth function piecewise defined by polynomials. Unfortunately, splines are not uniform functions and therefore, function parameters do not work either for the case of splines.

We will consider 2 growth curves. The first one is described by (

It is observed from Figures

Next, spline is calculated for each of the functions in the bunch illustrated in Figure

How can we apply

Suppose that the value vector of the

For each

Then (

After that we define a test statistic,

We can test the null hypothesis that the two groups of samples are not significantly different:

With the statistical framework of

Two bunches of growth curves of genotypes

Simulated organ growth curves of different genotypes

The relative measurement value of each sample at the beginning time point 0 is 0 percent, and that at the ending time point 20 is 100 percent. Consequently, we are not able to differentiate them merely by the measurement value at the mature state and have to resort to the difference of developmental processes.

Intuitively, the two groups are far apart. But we fail to apply the functional mapping framework to differentiate the two groups, since there exist no parameters like

But using the statistical framework given in Section

As is mentioned earlier, a curve may theoretically provide infinite amount of information about growth. Though the

Where and why is

The limitation of

Comparing the sizes of shade in Figures

This example indicates that

Suppose that

The

Suppose

It is easy to prove that growth dissimilarity satisfies distance axioms; that is,

And we hence have transformed problems about growth curves into problems about vectors which will help to analyze the relation between different growth curves in Figure

The results above show that, in the growth perspective of earlier and later halves,

Different weights can be designated to differently important phases of growth according to specific problems. So the growth dissimilarity defined in (

Equation (

More and more growth traits are available and they can be described by growth curves. Studies [

In order to identify genes with growth traits, we are required to divide into groups all growth curves that are as similar as possible in the same group while being as dissimilar as possible in different groups. But a common situation we are encountering is that it is difficult for us to obtain reasonable groups.

With the

Though describing rules are capable of grouping growth curves, human experts are involved in prescribing the rules, and thus the rules, consequently the grouping results, may be different from person to person.

Unlike the grouping technique with describing rules, the clustering technique,

We simulated the

Clustering growth curves based on their

In order to generalize functional mapping and overcome the shortages of it,

A function globally and thoroughly describing the process of growth is unnecessary for calculating

Being a key and general characteristic value though,

By extracting the growth information in a curve and forming a vector, we can use well developed techniques for analysis, such as describing rules and

The authors declare that there are no competing interests regarding the publication of this paper.

This study was supported by the Fundamental Research Funds for the Central Universities (YX-2010-30), NSFC Grant (31470675), and Special Fund for Forest Scientific Research in Public Welfare of China (201404102).