Group Title: Theoretical Biology and Medical Modelling 2007, 4:5
Title: A computational model for functional mapping of genes that regulate intra-cellular circadian rhythms
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Title: A computational model for functional mapping of genes that regulate intra-cellular circadian rhythms
Series Title: Theoretical Biology and Medical Modelling 2007, 4:5
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Creator: Liu T
Liu X
Chen Y
Wu R
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Theoretical Biology and Medical

Modelling


B
BioMed Central


Research


A computational model for functional mapping of genes that
regulate intra-cellular circadian rhythms
Tian Liu', Xueli Liu', Yunmei Chen2 and Rongling Wu*1


Address: 'Department of Statistics, University of Florida, Gainesville, FL 32611, USA and 2Department of Mathematics, University of Florida,
Gainesville, FL 32611, USA
Email: Tian Liu tianliu@stat.ufl.edu; Xueli Liu xueli@stat.ufl.edu; Yunmei Chen yun@math.ufl.edu; Rongling Wu* rwu@stat.ufl.edu
* Corresponding author


Published: 30 January 2007
Theoretical Biology and Medical Modelling 2007, 4:5 doi: 10.1 186/1742-4682-4-5
This article is available from: http://www.tbiomed.com/content/4/l/5


Received: 8 October 2006
Accepted: 30 January 2007


2007 Liu et al; licensee BioMed Central Ltd.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0),
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.



Abstract
Background: Genes that control circadian rhythms in organisms have been recognized, but have
been difficult to detect because circadian behavior comprises periodically dynamic traits and is
sensitive to environmental changes.
Method: We present a statistical model for mapping and characterizing specific genes or
quantitative trait loci (QTL) that affect variations in rhythmic responses. This model integrates a
system of differential equations into the framework for functional mapping, allowing hypotheses
about the interplay between genetic actions and periodic rhythms to be tested. A simulation
approach based on sustained circadian oscillations of the clock proteins and their mRNAs has been
designed to test the statistical properties of the model.
Conclusion: The model has significant implications for probing the molecular genetic mechanism
of rhythmic oscillations through the detection of the clock QTL throughout the genome.


Background
Rhythmic phenomena are considered to involve a mecha-
nism, ubiquitous among organisms populating the earth,
for responding to daily and seasonal changes resulting
from the planet's rotation and its orbit around the sun [1].
All these periodic responses are recorded in a circadian
clock that allows the organism to anticipate rhythmic
changes in the environment, thus equipping it with regu-
latory and adaptive machinery [2]. It is well recognized
that circadian rhythms operate at all levels of biological
organization, approximating a twenty-four hour period,
or more accurately, the alternation between day and night
[3]. Although there is a widely accepted view that the nor-
mal functions of biological processes are strongly corre-
lated with the genes that control them, the detailed


genetic mechanisms by which circadian behavior is gener-
ated and mediated are poorly understood [4].

Several studies have identified various so-called circadian
clock genes and clock-controlled transcription factors
through mutants in animal models [5,6]. These genes
have implications for clinical trials: their identification
holds great promise for determining optimal times for
drug administration based on an individual patient's
genetic makeup. It has been suggested that drug adminis-
tration at the appropriate body time can improve the out-
come of pharmacotherapy by maximizing the potency
and minimizing the toxicity of the drug [7], whereas drug
administration at an inappropriate body time can induce
more severe side effects [8]. In practice, body-time-
dependent therapy, termed chronotherapy [9], can be


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Theoretical Biology and Medical Modelling 2007, 4:5



optimized via the genes that control expression of the
patient's physiological variables during the course of a
day.

With the completion of the Human Genome Project, it
has been possible to draw a comprehensive picture of the
genetic control of the functions of the biological clock
and, ultimately, to integrate genetic information into rou-
tine clinical therapies for disease treatment and preven-
tion. To achieve this goal, there is a pressing need to
develop powerful statistical and computational algo-
rithms for detecting genes or quantitative trait loci that
determine circadian rhythms as complex dynamic traits.
Unlike many other traits, rhythmic oscillations are gener-
ated by complex cellular feedback processes comprising a
large number of variables. For this reason, mathematical
models and numerical simulations are needed to grasp
the molecular mechanisms and functions of biological
rhythms fully [10]. These mathematical models have
proved useful for investigating the dynamic bases of phys-
iological disorders related to perturbations of biological
behavior.

In this article, we will develop a statistical model for
genetic mapping of QTL that determine patterns of rhyth-
mic responses, using random samples from a natural pop-
ulation. This model is implemented by the principle of
functional mapping [11], a statistical framework for map-
ping dynamic QTL for the pattern of developmental
changes, by considering systems of differential equations
for biological clocks. Simulation studies have been per-
formed to investigate the statistical properties of the
model.

Model
Mathematical Modeling of Circadian Rhythms
In all organisms studied so far, circadian rhythms that
allow adaptation to a periodically changing environment
originate from negative autoregulation of gene expres-
sion. Scheper et al. [10] illustrated and analyzed the gen-
eration of a circadian rhythm as a process involving a
reaction cascade containing a loop, as depicted in Fig. 1A.
The reaction loop consists in the production of the effec-
tive protein from its mRNA and negative feedback from
the effective protein on mRNA production. The protein
production process involves translation and subsequent
processing steps such as phosphorylation, dimerization,
transport and nuclear entry. It is assumed that the protein
production cascade and the negative feedback are nonlin-
ear processes in the reaction loop (Fig. 1B), with a time
delay between protein production and subsequent
processing. These nonlinearities and the delay critically
determine the free-running periodicity in the feedback
loop.


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Scheper et al. [10] proposed a system of coupled differen-
tial equations to describe the circadian behavior of the
intracellular oscillator:



dM rM
d n qMM



dP
- = rpM(t -r)m qpP
dt

where M and P are, respectively, the relative concentra-
tions of mRNA and the effective protein measured at a
particular time, r, is the scaled mRNA production rate
constant, rp is the protein production rate constant, q4 and
qp are, respectively, the mRNA and protein degradation
rate constants, n is the Hill coefficient, m is the nonlinear
exponent in the protein production cascade, r is the total
duration of protein production from mRNA, and k is a
scaling constant.

Equation 1 constructs an unperturbed (free-running) sys-
tem of the intracellular circadian rhythm generator that is
defined by seven parameters, ,, = (n, m, r, rM, rp, q4, qp,
k). The behavior of this system can be determined and
predicted by changes in these parameter combinations.
For a given QTL, differences in the parameter combina-
tions among genotypes imply that this QTL is involved in
the regulation of circadian rhythms. Statistical models
will be developed to infer such genes from observed
molecular markers such as single nucleotide polymor-
phisms (SNPs).

Statistical Modeling of Functional Mapping
Suppose a random sample of size N is drawn from a nat-
ural human population at Hardy-Weinberg equilibrium.
In this sample, multiple SNP markers are genotyped, with
the aim of identifying QTL that affect circadian rhythms.
The relative concentrations of mRNA (M) and the effective
protein (P) are measured in each subject at a series of time
points (1, ..., T), during a daily light-dark cycle. Thus,
there are two sets of serial measurements, expressed as
[M(1), ..., M(T)l and [P(1),..., P(T)]. According to the dif-
ferential functions (1), these two variables, modeled in
terms of their change rates, are expressed as differences
between two adjacent times, symbolized by

y= [M(2)- M(1),..., M(T)- M(T- 1)1

= [y(l),..., y(T 1)

for the protein change and

z= [P(2)- P(1)..., P(T)- P(T- 1)1


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Theoretical Biology and Medical Modelling 2007, 4:5


SCN Pacemaker Neuron


N,


AN 4W 406


PROTEIN
ocess ing


i i j

Pr-IT



PROTEIN* --

Circadian
rhythm
expression


I


I

I
I

I
I


non
linear






4I ft


o rr


mRNA q

I degradation


delay





non
linear


rP


PROTEIN*
degradation


Figure I
(A) Diagram of the biological elements of the protein synthesis cascade for a circadian rhythm generator. (B) Model interpreta-
tion of A showing the delay (z) and nonlinearity in the protein production cascade, the nonlinear negative feedback, and mRNA
and protein production (rM, rp) and degradation (qm, qp). Adapted from ref. [10].

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Theoretical Biology and Medical Modelling 2007, 4:5


[z(1)..., z(T- 1)1


for the mRNA change.

Assume that a putative QTL with alleles A and a affecting
circadian rhythms is segregated in the population. The fre-
quencies of alleles A and a are q and 1 q, respectively. For
a particular genotype j of this QTL (j = 0 for aa, 1 for Aa
and 2 for AA), the parameters describing circadian
rhythms are denoted by ,i = (ni, my i,, rM, rpj, q, qpi, kj).
Comparisons of these quantitative genetic parameters
among the three different genotypes can determine
whether and how this putative QTL affects circadian
rhythms.

The time-dependent phenotypic changes in mRNA and
protein traits for individual i measured at time t due to the
QTL can be expressed by a bivariate linear statistical
model


2
yi(I) = um(t)+ e (t)
j=o
2
z(t)= I jupO(t) + e(t)
j=0


where (ij is an indicator variable for the possible geno-
types of the QTL for individual i, defined as 1 if a particu-
lar QTL genotypej is indicated and 0 otherwise, UMj(t) and
upj(t) are the genotypic values of the QTL for mRNA and
protein changes at time t, respectively, which can be deter-
mined using the differential functions expressed in equa-
tion (1), and eY (t) and ef (t) are the residual effects in
individual i at time t, including the aggregate effect of
polygenes and error effects.

The dynamic features of the residual errors of these two
traits can be described by the antedependence model,
originally proposed by Gabriel [12] and now used to
model the structure of a covariance matrix [13]. This
model states that an observation at a particular time t
depends on the previous ones, the degree of dependence
decaying with time lag. Assuming the st-order structured
antedependence (SAD(l)) model, the relationship
between the residual errors of the two traits y and z at time
t for individual i can be modeled by


e~(t) = eY (t 1) + yvze (t 1) + e (t)
e =e t-(3)
ef (t) = Oef (t 1) + pze (t 1) + ef (t)


where 4 and Vlk are, respectively, the antedependence
parameters caused by trait k itself and by the other trait,
and ei (t) and ef (t) are the time-dependent innovation
error terms, assumed to be bivariate normally distributed
with mean zero and variance matrix


, (t)y (t)p(t)

82 (t) J


where 85 (t) and 85 (t) are termed time-dependent inno-
vation variances. These variances can be described by a
parametric function such as a polynomial of time [14],
but are assumed to be constant in this study. p(t) is the
correlation between the error terms of the two traits, spec-
ified by an exponential function of time t [14], but is
assumed to be time-invariant for this study. It is reasona-
ble to say that there is no correlation between the error
terms of two traits at different time points, i.e.
Corr( e (ty), ef (t)) = 0 (ty tJ.

Based on the above conditions, the covariance matrix (E)
of phenotypic values for traits y and z can be structured in
terms of ,, 0, V/, Vy / and Z,(t) by a bivariate SAD(1)
model [15,16]. Also, the closed forms for the determinant
and inverse of Y can be derived as given in [15,161. We use
a vector of parameters arrayed in , = (' ,, /, 6', 6,
p) to model the structure of the covariance matrix
involved in the function mapping model.

Likelihood
The likelihood of samples with 2(T 1)-dimensional meas-
urements, x = (xi)= {y(t),zi(t)}t=l, for individual i and
marker information, M, in the human population affected
by the QTL is formulated on the basis of the mixture
model, expressed as


L(co, x,M)= [ Ojlifj(xi;eu.,Pv)1 (4)

where the unknown parameters include two parts, a =
(aj i) and = (,i 0,). In the statistics, the parameters a
= ()ji) determine the proportions of different mixture
normals, and actually reflect the segregation of the QTL in
the population, which can be inferred on the basis of non-
random association between the QTL and the markers.
For a mapping population, N progeny can be classified
into different groups on the basis of known marker geno-
types. Thus, in each such marker genotype group, the mix-
ture proportions of QTL genotypes (i, i) can be expressed

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z)2 (t)
Y1 (t) =
,x(t)Sy(t)p(t)







Theoretical Biology and Medical Modelling 2007, 4:5



as the conditional probability of QTL genotype j for sub-
ject i given its marker genotype.

Suppose that this QTL is genetically associated with a
codominant SNP marker that has three genotypes, MM,
Mm and mm. Let p and 1 p be the allele frequencies of
marker alleles M and m, respectively, and D be the coeffi-
cient of (gametic) linkage disequilibrium between the
marker and QTL. According to linkage disequilibrium-
based mapping theory [17], the detection of significant
linkage disequilibrium between the marker and QTL
implies that the QTL may be linked with and, therefore,
can be genetically manipulated by the marker. The four
haplotypes for the marker and QTL are MA, Ma, mA and
ma, with respective frequencies expressed as pll = pq + D,
lo = p(1 4) D, Po0 = (1 p)q D and oo = (1 p)(1 q) +
D. Thus, the population genetic parameters p, q, D can be
estimated by solving a group of regular equations if we
can estimate the four haplotype frequencies. The condi-
tional probabilities of QTL genotypes given marker geno-
types in a natural population can be expressed in terms of
the haplotype frequencies (see [18]).


In the mixture model (4), 0= {(Ou,,O) } o0 is the
unknown vector that determines the parametric family f,
described by a multivariate normal distribution with the
genotype-specific mean vector

u,= (UM; Up )= {UM(t);Up(t) 1

r -M qMJM(t + l);rpJM(t+ rJ) qpjP(t + ) (5)
SP(t+ 1) f t1

and the covariance matrix Y. While the mean vector is
determinedby genotype-specific parameters, ,j = (nj, mj,
,' rAj, rpj, q4j, qPj, k), j = (2,1,0) the covariance matrix is
structured by common parameters, 0, = ('0, 0,, Vu, r,, ,
4, p).

Algorithm
Wang and Wu [18] proposed a closed form for the EM
algorithm to obtain the maximum likelihood estimates
(MLEs) of haplotype frequencies p11, plo, po, and poo, and
thus the allele frequencies of the marker (p) and QTL (q)
and their linkage disequilibrium (D). Genotype-specific
mathematical parameters in uj (5) for the two differential
functions of circadian rhythms, and the parameters that
specify the structure of the covariance matrix, Y, can be
theoretically estimated by implementing the EM algo-
rithm. But it would be difficult to derive the log-likeli-
hood equations for these parameters by this approach
because they are related in a complicated nonlinear way.
The simplex algorithm, which relies only upon a target


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function, has proved powerful for estimating the MLEs of
these parameters [19] and will be used in this study. As
discussed above, closed forms exist for the determinant
and inverse and should be incorporated into the estima-
tion process to increase computational efficiency.

Hypothesis Testing
One of the most significant advantages of functional map-
ping is that it can ask and address biologically meaningful
questions about the interplay between gene actions and
trait dynamics by formulating a series of hypothesis tests.
Wu et al. [20] described several general hypothesis tests
for different purposes. Although all these general tests can
be used directly in this study, we propose here the most
important and specific tests for the existence of QTL that
affect mRNA and protein changes pleiotropically or sepa-
rately, and for the effects of the QTL on the shape of dif-
ferential functions.

Existence of QTL
Testing whether a specific QTL is associated with the dif-
ferential functions (1) is a first step toward understanding
the genetic architecture of circadian rhythms. The genetic
control of the entire rhythmic process can be tested by for-
mulating the following hypotheses:

Ho: D=0 vs. HI: D 0 (6)

H0 states that there are no QTL affecting circadian rhythms
(the reduced model), whereas H1 proposes that such QTL
do exist (the full model). The statistic for testing these
hypotheses (6) is calculated as the log-likelihood ratio
(LR) of the reduced to the full model:


LR1 = -2[ln L(O, Ix, M)- InL(O, & Ix, M)], (7)

where the tildes and hats denote the MLEs of the
unknown parameters under Ho and H1, respectively. The
LR is asymptotically X2-distributed with one degree of
freedom.

A similar test for the existence of a QTL can be performed
on the basis of these hypotheses, as follows:

Ho :j 0,,, j = (2,1,0) (8)

H : At least one of the qualities above does not hold;

from which the LR is calculated by


LR2: -2[ln L(9 Ix) In(0, I|x, M)], (9)

with the doubled tildes denoting the estimates under Ho
of hypothesis (8). It is difficult to determine the distribu-


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tion of the LR2 because the linkage disequilibrium is not
identifiable under H1. An empirical approach to deter-
mining the critical threshold is based on permutation
tests, as advocated by Churchill and Doerge [21]. By
repeatedly shuffling the relationships between marker
genotypes and phenotypes, a series of maximum LR2 val-
ues are calculated, from the distribution of which the crit-
ical threshold is determined.

Is the QTL for mRNA or protein rhythms?
After the existence of a QTL that affects circadian rhythms
is confirmed, we need to test whether it affects the rhyth-
mic responses of mRNA and protein jointly or separately.
The hypothesis for testing the effect of the QTL on the
mRNA response is formulated as

Ho: (rmj, q1, kh, n,) (rM, q4, k, n) forj =0, 1, 2 (10)

H1 : At least one of the qualities above does not hold.

The log-likelihood values under Ho and H, are calculated,
and thus the corresponding LR.

A similar test is formulated for detecting the effect of the
QTL on the protein rhythm:

Ho: (rpj, qpj, rj, mj,) (rp, qp, T, m) for j = 0, 1, 2 (11)

Hi : At least one of the qualities above does not hold.

For both hypotheses (10) and (11), an empirical
approach to determining the critical threshold is based on
simulation studies. If the null hypotheses of (10) and (11)
are both rejected, this means that the QTL exerts a pleio-
tropic effect on the circadian rhythms of mRNA and pro-
tein.

The QTL responsible for the behavior and shape of
circadian rhythms
Two different subspaces of parameters are used to define
the features of circadian rhythms: {n, m, r}, determining
the nonlinearity and delay in the system, and {rM, rp, q,,
qp}, determining the phase-response curves. The null
hypotheses regarding the genetic control of the system's
oscillatory behavior and the shape of the rhythmic
responses are:

Ho : (nj,m,Tj) -(n,m,T)
H0 (r, rp,qM,qj) (M,rp,qM,q) forj= 0, 1, 2 (12)

The oscillatory behavior of a circadian rhythm can also be
determined by the amplitude of the rhythm, defined as
the difference between the peak and trough values; its
phase, defined as the timing of a reference point in the
cycle (e.g. the peak) relative to a fixed event (e.g. begin-


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ning of the night phase); and its period, defined as the
time interval between phase reference points (e.g. two
peaks). The genetic determination of all thesevariables
can be tested.

Simulation
Simulation experiments are performed to examine the sta-
tistical properties of the model proposed for genetic map-
ping of circadian rhythms. We choose 200 individuals at
random from a human population at Hardy-Weinberg
equilibrium. Consider one of the markers genotyped for
all subjects. This marker, with two alleles M and m, is used
to infer a QTL with two alleles A and a for circadian
rhythms on the basis of non-random association. The
allele frequencies are assumed to be p = 0.6 for allele M
and q = 0.6 for allele A. A positive value of linkage disequi-
librium (D = 0.08) between M and A is assumed, suggest-
ing that these two more common alleles are in coupled
phase [22].

The three QTL genotypes, AA, Aa and aa, are each hypoth-
esized to have different response curves for circadian
rhythms of mRNA and protein as described by equation
(1). The rhythmic parameters ,(j = (nj, im, yj, rmy, rpj, qmj,
qpj, k) for the three genotypes, given in Table 1, are deter-
mined in the ranges of empirical estimates of these param-
eters [10]. Note that for computational simplicity the
scaling constant k and the total duration of protein pro-
duction from mRNA are given values 1 and 4.0, respec-
tively. We used the SAD(1) model to structure the
covariance matrix based on the antedependence parame-
ters (0,, zy, V/<, qy) and innovation variances (82, 8 )
(Table 1). The innovation variances for each of the two
rhythmic traits were determined by adjusting the herita-
bility of the curves to H2 = 0.1 and 0.4, respectively, due to
the QTL for the rhythmic response at a middle measure-
ment point.

Many factors have been shown to affect the precision of
parameter estimation and the power of QTL detection by
functional mapping. These factors are related to experi-
mental design (sample size and number and pattern of
repeated measures), the genetic properties of the circadian
rhythm heritabilityy of the curves, population genetic
parameters of the underlying QTL), and the analytical
approach to modeling the structure of the covariance
matrix. Previous studies have investigated the properties
of functional mapping when different experimental
designs are used [15,18]. For this simulation study, we
focus on the influence of different heritabilities on param-
eter estimation using a practically reasonable sample size
(n = 200). We assumed that the relative concentrations of


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Table I: The MLEs of parameters that define circadian rhythms for three different QTL genotypes, the structure of the covariance
matrix and the association between the marker and QTL in a natural population, taking the heritability of the assumed QTL as H2 =
0.I. The numbers in parentheses are the square roots of the mean square errors of the MLEs.


Rhythmic Parameters


MLE

1.57(0.067)
3.02(0.018)
1.16(0.060)
0.30(0.020)
0.15(0.008)
0.16(0.001)


Given

1.40
2.90
1.30
0.35
0.17
0.17


Matrix Structuring Parameters


Given MLE


0.011(0.001)
0.098(0.001)
0.105(0.005)
-0.206(0.006)
0.223(0.001)


1.742(0.100)


0.216(0.016)


Genetic Parameters


Given MLE


0.601(0.003)
0.501(0.094)
0.068(0.012)


mRNA and protein are measured at eight equally-spaced
time points in each subject, although these measurements
can be made differently in terms of the number and pat-
tern of repeated measures.

The phenotypic values of circadian rhythms for the mRNA
and protein traits are simulated by summing the geno-
typic values predicted by the rhythmic curves and residual
errors following a multivariate normal distribution, with
MVN(0, Y). The simulated phenotypic and marker data
were analyzed by the proposed model. The population
genetic parameters of the QTL can be estimated with rea-
sonably high precision using a closed-form solution
approach [18]. We compare the estimation of the marker
allele frequencies, QTL allele frequencies and marker-QTL
linkage disequilibria under different heritability levels.


The precision of estimation of marker allele frequency is
not affected by differences in heritability, but estimates of
QTL allele frequency and marker-QTL linkage disequilib-
rium are more precise for a higher (Table 1) than a lower
(Table 2) heritability.

Figure 2A illustrates different forms of circadian rhythms
for three QTL genotypes, AA, Aa and aa, with the rhythmic
values for the protein and mRNA responses given in
Tables 1 and 2. Pronounced differences among the geno-
types imply that the QTL may affect the joint rhythmic
response of the protein and mRNA concentrations. The
rhythmic values can be estimated reasonably from the
model. Using the estimates of the rhythmic parameters
from one random simulation, we draw the oscillations of
the two traits. The shapes of these curves seem to be



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Given


MLE

1.54(0.139)
2.91(0.010)
1.35(0.054)
0.36(0.007)
0.16(0.011)
0.16(0.005)


Given

1.70
2.80
0.90
0.40
0.18
0.18


MLE

1.74(0.045)
2.79(0.050)
0.98(0.083)
0.39(0.001)
0.17(0.012)
0.18(0.001)


0.010
-0.100
0.100
-0.200
0.223


0.200


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Theoretical Biology and Medical Modelling 2007, 4:5


Table 2: The MLEs of parameters that define circadian rhythms for three different QTL genotypes, the structure of the covariance
matrix and the association between the marker and QTL in a natural population, taking the heritability of the assumed QTL as H2 =
0.4. The numbers in parentheses are the square roots of the mean square errors of the MLEs.


Rhythmic Parameters


MLE

1.52(0.059)
3.02(0.015)
1.14(0.052)
0.30(0.009)
0.16(0.003)
0.16(0.004)


Given

1.40
2.90
1.30
0.35
0.17
0.17


Matrix Structuring Parameters


Given MLE


0.010(0.001)
0.095(0.002)
0.102(0.005)
0.201(0.006)
0.309(0.01 1)


0.204(0.011)


0.038(0.002)


Genetic Parameters


Given MLE


0.601(0.002)
0.67(0.091)
0.067(0.022)


broadly consistent with those of the hypothesized curves,
although the curve estimates are more accurate under
higher (Fig. 2C) than lower (Fig. 2B) heritability.

The estimates of the rhythmic parameters for each
response curve also display reasonable precision, as
assessed by the square roots of the mean square errors
over 100 repeated simulations. As expected, the estimate
is more precise when the heritability increases from 0.1
(Table 1) to 0.4 (Table 2). The model displays great power
in detecting a QTL responsible for circadian rhythms
using the marker associated with it. Given the above sim-
ulation conditions, a significant QTL can be detected with
about 75% power for a heritability of 0.1. The power
increases to over 90% as the heritability increases to 0.4.


The model can be used to test whether the QTL detected
for overall protein and mRNA rhythm responses also
affects key features of circadian rhythms, such as period,
amplitude or phase shift, by formulating the correspond-
ing hypotheses. For a real data set, it is exciting to test
these hypotheses because they may enable the mechanis-
tic basis of the genetic regulation of circadian rhythms to
be identified. In the current simulation, these hypothesis
tests were not performed.

Discussion
One of the most important aspects of life is the rhythmic
behavior that is rooted in the many regulatory mecha-
nisms that control the dynamics of living systems. The
most common biological rhythms are circadian rhythms,



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Given


MLE

1.42(0.006)
2.90(0.009)
1.34(0.028)
0.35(0.002)
0.17(0.011)
0.17(0.003)


Given

1.70
2.80
0.90
0.40
0.18
0.18


MLE

1.71(0.060)
2.80(0.020)
0.93(0.066)
0.40(0.001)
0.18(0.005)
0.18(0.001)


0.010
-0.100
0.100
-0.200
0.307


0.200


0.037


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Theoretical Biology and Medical Modelling 2007, 4:5


5


5 -~


1 1.5 2


1 1.5 2


1 1.5 2


1 1.5 2


31-


Figure 2
Free-running oscillation of mRNA abundance (x) and protein abundance (y) in a rhythmic system, expressed as limit cycle con-
tour, annotated with the time points within the 24.6 h circadian cycle, for three assumed QTL genotypes using given rhythmic
parameter values (A), estimated values under H2 = 0.1 (B), and estimated values under H2 = 0.4 (C). The three plots within
each column correspond to QTL genotypes AA, Aa and aa, respectively.


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Theoretical Biology and Medical Modelling 2007, 4:5



which occur with a period close to 24 h, allowing organ-
isms to adapt to periodic changes in the terrestrial envi-
ronment [1]. With the rapid accumulation of new data on
gene, protein and cellular networks, it is becoming
increasingly clear that genes are heavily involved in the
cellular regulatory interactions underpinning circadian
rhythms [4,23]. However, a detailed picture of the genetic
architecture of circadian rhythms has not been obtained,
although ongoing projects such as the Human Genome
Project will assist in the characterization of circadian
genetics.

Traditional strategies for identifying circadian clock genes
in mammals have been based on the analysis of single
gene mutations and the characterization of genes identi-
fied by cross-species homology, and have laid an essential
groundwork for circadian genetics [6,23]. However, these
strategies do not include a more thorough examination of
the breadth and complexity of influences on circadian
behavior throughout the entire genome. Genetic mapping
relying upon genetic linkage maps has provided a power-
ful tool for identifying the quantitative trait loci (QTL)
responsible for circadian rhythms. In a mapping study of
196 F2 hybrid mice, Shimomura et al. [24] detected 14
interacting QTL that contribute to the variation of rhyth-
mic behavior in mice by analyzing different discrete
aspects of circadian behavior: free-running circadian
period, phase angle of entrainment, amplitude of the cir-
cadian rhythm, circadian activity level, and dissociation of
rhythmicity.

The data of Shimomura et al. [24] point to promising
approaches for genome-wide analysis of rhythmic pheno-
types in mammals including humans. Their most signifi-
cant drawback is the lack of robust statistical inferences
about the dynamic genetic control of circadian rhythms.
Typically, biological rhythms are dynamic traits, and the
pattern of their genetic determination can change dramat-
ically with time. In this article, we have incorporated
mathematical models and concepts regarding the molec-
ular and cellular mechanisms of circadian rhythms into a
general framework for mapping dynamic traits, called
functional mapping [11]. Based firmly on experiments,
robust differential equations have been established to
provide an essential tool for studying and comprehending
the cellular networks for circadian rhythms [1,25-27]. As
an attempt to integrate differential equations into func-
tional mapping, the statistical model shows favorable
properties in estimating the effects of a putative QTL and
its association with polymorphic markers. The simulation
study results suggest that the parameters determining the
behavior and shape of circadian rhythmic curves can be
estimated reasonably even if the QTL effect is small to
moderate. As seen in general functional mapping [11], the
model implemented with a system of differential equa-


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tions also allows us to make a number of biologically
meaningful hypothesis tests for understanding the genetic
control of rhythmic responses in organisms.

As a first attempt of its kind, the model proposed in this
article has only considered one QTL associated with circa-
dian rhythms. A one-QTL model is definitely not suffi-
cient to explain the complexity of the genetic control of
this trait. A model incorporating multiple QTL and their
interactive networks should be derived; this is technically
straightforward. In addition, the system of circadian
rhythms is characterized by two variables, and this may
also be too simple to reflect the complexity of rhythmic
behavior. A number of more sophisticated models, gov-
erned by systems of five [28], ten [29] or 16 kinetic equa-
tions [4,30,31], have been constructed to describe the
detailed features of a rhythmic system in regard to
responses to various internal and environmental factors.
While the identification of circadian clock genes can elu-
cidate the molecular mechanism of the clock, our model
will certainly prove its value in elucidating the genetic
architecture of circadian rhythms and will probably lead
to the detection of the driving forces behind circadian
genetics and its relationship to the organism as a whole.

Competing interests
The authors) declare that they have no competing inter-
ests.

Authors' contributions
The entire theoretical concept of the work was envisaged
by RLW. The mathematical and statistical modeling was
carried out by TL with feedback from XLL and YMC.

Acknowledgements
The preparation of this manuscript was supported by NSF grant (0540745)
to RLW.

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USA 2003, 100:7051-7056. Your research papers will be:

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