Equilibrium processes in biomedical data analysis: the Wright–Fisher model
The biological process of cooperative interaction with equilibrium state will be described as a model of binary statistical experiments with Wright–Fisher normalization, which sets the concentration of a certain characteristic. Such mathematical model is supposed to have a property of persistent reg...
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Інститут кібернетики ім. В.М. Глушкова НАН України
2014
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irk-123456789-1247432017-10-04T03:02:46Z Equilibrium processes in biomedical data analysis: the Wright–Fisher model Koroliouk, D. Koroliuk, V.S. Rosato, N. Системный анализ The biological process of cooperative interaction with equilibrium state will be described as a model of binary statistical experiments with Wright–Fisher normalization, which sets the concentration of a certain characteristic. Such mathematical model is supposed to have a property of persistent regression which means that all current elementary transitions depend on the mean concentration of the said characteristics in the previous state. Equilibrium state of the model is expressed in the terms of the regression function, given by a cubic parabola with three real roots. We construct stochastic approximation of the model by autoregressive process with normal disturbances. Such approach was developed for effective and calculable mathematical description of dynamic concentration for experiment planning, parameters evaluation and hypotheses verification of mechanism of action. Біологічний процес кооперативної взаємодії з рівноважним станом описано як модель бінарних статистичних експериментів з нормалізацією Райта–Фішера, що встановлює концентрацію певної ознаки. Така математична модель, як передбачається, має властивість наполегливої регресії, яка означає, що всі поточні елементарні переходи залежать від середньої концентрації зазначеної ознаки в попередньому стані. Рівноважний стан моделі виражено в термінах функції регресії у вигляді кубічної параболи з трьома дійсними коренями. Побудовано стохастичну апроксимацію моделі процесом авторегресії з нормальними збуреннями. Такий підхід розроблено для конструктивного математичного опису динамічної концентрації для планування експерименту, оцінки параметрів і перевірки гіпотез механізму дії. 2014 Article Equilibrium processes in biomedical data analysis: the Wright–Fisher model / D. Koroliouk, V.S. Koroliuk, N. Rosato // Кибернетика и системный анализ. — 2014. — Т. 50, № 6. — С. 80-86. — Бібліогр.: 6 назв. — англ. 0023-1274 http://dspace.nbuv.gov.ua/handle/123456789/124743 519.24 en Кибернетика и системный анализ Інститут кібернетики ім. В.М. Глушкова НАН України |
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Системный анализ Системный анализ |
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Системный анализ Системный анализ Koroliouk, D. Koroliuk, V.S. Rosato, N. Equilibrium processes in biomedical data analysis: the Wright–Fisher model Кибернетика и системный анализ |
| description |
The biological process of cooperative interaction with equilibrium state will be described as a model of binary statistical experiments with Wright–Fisher normalization, which sets the concentration of a certain characteristic. Such mathematical model is supposed to have a property of persistent regression which means that all current elementary transitions depend on the mean concentration of the said characteristics in the previous state. Equilibrium state of the model is expressed in the terms of the regression function, given by a cubic parabola with three real roots. We construct stochastic approximation of the model by autoregressive process with normal disturbances. Such approach was developed for effective and calculable mathematical description of dynamic concentration for experiment planning, parameters evaluation and hypotheses verification of mechanism of action. |
| format |
Article |
| author |
Koroliouk, D. Koroliuk, V.S. Rosato, N. |
| author_facet |
Koroliouk, D. Koroliuk, V.S. Rosato, N. |
| author_sort |
Koroliouk, D. |
| title |
Equilibrium processes in biomedical data analysis: the Wright–Fisher model |
| title_short |
Equilibrium processes in biomedical data analysis: the Wright–Fisher model |
| title_full |
Equilibrium processes in biomedical data analysis: the Wright–Fisher model |
| title_fullStr |
Equilibrium processes in biomedical data analysis: the Wright–Fisher model |
| title_full_unstemmed |
Equilibrium processes in biomedical data analysis: the Wright–Fisher model |
| title_sort |
equilibrium processes in biomedical data analysis: the wright–fisher model |
| publisher |
Інститут кібернетики ім. В.М. Глушкова НАН України |
| publishDate |
2014 |
| topic_facet |
Системный анализ |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/124743 |
| citation_txt |
Equilibrium processes in biomedical data analysis: the Wright–Fisher model / D. Koroliouk, V.S. Koroliuk, N. Rosato // Кибернетика и системный анализ. — 2014. — Т. 50, № 6. — С. 80-86. — Бібліогр.: 6 назв. — англ. |
| series |
Кибернетика и системный анализ |
| work_keys_str_mv |
AT korolioukd equilibriumprocessesinbiomedicaldataanalysisthewrightfishermodel AT koroliukvs equilibriumprocessesinbiomedicaldataanalysisthewrightfishermodel AT rosaton equilibriumprocessesinbiomedicaldataanalysisthewrightfishermodel |
| first_indexed |
2025-07-09T01:57:55Z |
| last_indexed |
2025-07-09T01:57:55Z |
| _version_ |
1837132704716423168 |
| fulltext |
UDC 519.24
D. KOROLIOUK, V.S. KOROLIUK, N. ROSATO
EQUILIBRIUM PROCESSES IN BIOMEDICAL DATA ANALYSIS:
THE WRIGHT–FISHER MODEL
Abstract. The biological process of cooperative interaction with equilibrium state will be
described as a model of binary statistical experiments with Wright–Fisher normalization, which
sets the concentration of a certain characteristic. Such mathematical model is supposed to have
a property of persistent regression which means that all current elementary transitions depend on
the mean concentration of the said characteristics in the previous state. Equilibrium state of the
model is expressed in the terms of the regression function, given by a cubic parabola with three
real roots. We construct stochastic approximation of the model by autoregressive process with
normal disturbances. Such approach was developed for effective and calculable mathematical
description of dynamic concentration for experiment planning, parameters evaluation and
hypotheses verification of mechanism of action.
Keywords: binary statistical experiment, persistent regression, Wright–Fisher normalization,
equilibrium state, normal autoregression.
INTRODUCTION
In biological processes with equilibrium, the dynamics of concentration, or
frequencies, of a predefined characteristic, can be described by mathematical model
of binary statistical experiments, based on statistical data of elementary hypotheses
validation about the presence or absence of a predefined attribute A in the set of
elements that make up a complex system.
It is assumed that:
1) all the elements that make up the system can gain or lose the said attribute A
over time, that is the frequency of attribute A is dynamic variable;
2) the basic object of our study are statistical experiments, characterized by
relative frequencies of presence or absence of the attribute A in a sample of fixed
volume at each time instant;
3) it is assumed the dependency of (average) results of next experiment (at time
instant k �1) on average result of the present experiment at time instant k. This
relationship is called the feature of persistent regression and used as fundamental
condition for the subsequent analysis of the model.
The method of constructing and exploring of the proposed mathematical model is
based on analysis of the following basic properties of statistical experiments:
1. Persistent regression.
2. Equilibrium value and fluctuations, as well as their asymptotic behavior.
3. Approximation by normal process of autoregression.
Note that in view of assumption 2), the value of statistical experiment tends to
probability of the presence or absence of attribute A, by N � � , according to the law of
large numbers. However, such probability, according to assumption 3), has a complex
relationship and can be constructively expressed in terms of regression function.
As a result, all static and dynamic characteristics of the model can be expressed in
terms of the set of regression functions, or their transformations.
EXAMPLE: A MODEL OF BIOLOGICAL MOLECULES INTERACTION
Equilibrium processes are common in chemical and biochemical systems and play
important role in many mechanisms of interaction and self-regulation.
In studies of these processes, one should develop an adequate method of
description, analysis and prediction of the behavior of such systems, taking into
account the actions of a wide variety of external factors (for example, [1]).
80 ISSN 0023-1274. Êèáåðíåòèêà è ñèñòåìíûé àíàëèç, 2014, òîì 50, ¹ 6
© D. Koroliouk, V.S. Koroliuk, N. Rosato, 2014
As an example, we can refer to the enzyme Glutathione Transferase (GST) that
binds the tripeptide glutathione (GSH) as a substrate for subsequent organic reactions.
The GST is a very important enzyme in cellular biology, as a matter of fact it is
involved in many detoxifying reactions and in mechanism of drug resistance in cancer
cells [2]. This is a typical case where the stochastic nature of biological molecules
interaction is interpreted as a model of statistical experiments with regression function,
which sets the concentration S k
N
� ( ) of biological molecules bound to a ligand (enzymes
plus ligand), and of free biological molecules (enzyme without ligand) S kN
� ( ) at each
stage k of experiment. In this case N records the total number of bounded and free
molecules in biological experiments. The probabilities of transition to capture or release
of the ligand depend on the mean concentration of free and bound ligands in the previous
step. The biological process of capture and release of ligands by enzymes will be
described by a model of statistical experiments with Wright–Fisher normalization.
The choice of this model is caused by similarity of the process of biological molecules
interaction and the process of arrangement of two attributes in the loci.
Such mathematical model describes the dynamic process of interaction between
ligands and enzymes. Under this model, one can analize experimental data for
estimation of parameters which define its dynamics, verify the model adequacy,
exercise fitting of the model, develop the protocols of measurement procedures etc.
STATISTICAL MODEL FORMULATION
The starting point for the mathematical model construction is Wright–Fisher model
in population genetics which is used by more than seventy years.
Basing on the definition formulated in [3], we restrict ourselves to the case of two
genetic attributes A1 and A2 . This narrows our analysis on a special case which,
however, is very rich for numerous applications.
Consider statistical model with pairwise arrangement of such attributes at
particular locus for N representatives of the population. There are three ways of
attributes pairing: A A1 1, A A1 2 or A A2 1 and A A2 2 .
In order to study the model dynamics, let us denote p the frequency of attribute
A1, and q the frequency of attribute A2 , observed in previous experiment.
For the next experiment, the attribute A1 frequency we denote P� , and
the attribute A2 frequency we denote P� . As already noted, the probabilities P�
depend on the average value of experiment in previous time instant.
We will consider the Wright–Fisher model as a series of statistical experiments (SE),
defined by the amounts of sample values � �( ) ( ( )k kr� , 1� �r N ) , k 0 :
S k
N
k kN
r
N
r( ) ( ),�
�
1
0
1
� , (1)
in which the random variables �r k( ), 1� �r N , k 0, are equally distributed and
independent for each fixed k 0 which take binary values �1.
Thus, the values of SE, determined by (1), mean that the following equation takes
place:
S k S k S kN N N( ) ( ) ( )� �� � . (2)
Here S k
N
�
( ) means the relative frequency, of two values �1 and �1 in the sample:
S k
N
I k S k
N
I k
N
r
N
r N
r
N
r
�
�
�
�
� � � � � �
( ) ( ) , ( ) ( )
1
1
1
1
1 1
{ } {� � }, k 0 , (3)
where I A( ) is indicator of event A:
I A
A
A
( ) �
�
�
1
0
if event occurs;
if event does not occurs.
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The frequencies S k
N
�
( ) are uniquely determined by the values S kN ( ) in
the following way:
S k S k S k S k
N N N N
� �� � � �( ) [ ( )] / , ( ) [ ( )] /1 2 1 2
with obvious complete probability relation
S k S k
N N
� �� �( ) ( ) 1.
The sampling binary values are defined by conditional probabilities
P k S k s P s kr N{ }� ( ) | ( ) ( ),� � � � � �1 1 0. (4)
We shall consider the regression function P s� ( ) expressed in terms of Wright–Fisher
proportions [3, Ch. 10, p. 412]:
P s W p q W p q� ��( ) : ( , ) / ( , ) ,
W p q p W p q W p q q W q p� �� � � �( , ) : ( ), ( , ) : ( )1 2 , (5)
W p q W p q W p q W p pq W q p q( , ) : ( , ) ( , ) ,� � � � � � �� � 1
2
2
22 1,
where the viability parameters W W1 2, are positive constants.
In this case, the values p q p, ,� �1 and s are connected as follows:
p s q s s p q s: ( ) / , : ( ) / , , | |� � � � � � �1 2 1 2 1. (6)
The initial suppositions (1)–(5) imply that the sequence of SE (1) has the property
of persistent regression:
E S k S k s C s s kN N[ ( ) | ( ) ] ( ), | | ,� � � � 1 1 0 , (7)
in which the regression function with Wright–Fisher normalization has
the following form (see [3, Ch. 10]):
C s W s W s W s( ) [ ( ) ( )] / ( )� �� � . (8)
TRANSFORMATION OF REGRESSION FUNCTIONS
Asymptotic analysis (as N � �) of statistic experiments (1)–(5) and their
increments involves the use of the corresponding regression functions or their
transformations.
Introducing new notations for reasons of symmetry, we can express the functions
in (8) using viability parameters:
V W V W p p p q p� � � �� � � � � � � �: , : ; : , : ,1 1 11 2
W p p p V W p p p V� � � � � � � � � �� � � �( ) ( ), ( ) ( ),1 1
p p p s p p p s: ( , ); , ( ) / ,� � � � �� � � � � 1 2 (9)
W p V p V p( ) [ ],� � �� � � �1 2 2
W s V s V s( ) [ ( ) ( ) ] / .� � � � �� �1 1 1 42 2
Now consider the process of increments of statistical experiments (1)–(3):
� �S k S k S k S k S k S k
N N N N N N
� � �� � � � � �( ) : ( ) ( ), ( ) : ( ) ( )1 1 .
Introduce regression functions with Wright–Fisher normalization of increments
C s0 ( ) and C p
0
�
( ):
E S k S k s C s W s
E S k S k p C
N N
N N
[ ( ) | ( ) ] ( ) / ( ),
[ ( ) | ( ) ]
�
�
� �
� �� �
0
0
�
( ) / ( ).p W p
(10)
82 ISSN 0023-1274. Êèáåðíåòèêà è ñèñòåìíûé àíàëèç, 2014, òîì 50, ¹ 6
The corresponding regression functions are denoted as:
C s E S k S k s C p E S k S kN N N N
( ) : [ ( ) | ( ) ], ( ) : [ ( ) | ( )� � � � � �� � �
1 1 p� ]. (11)
So one has
C s s C s W s C p p C p W p( ) ( ) / ( ); ( ) ( ) / ( )� � � ��
�
�
0 0
. (12)
Hereinafter, the regression function with Wright–Fisher normalization will be used
in the following modified form.
Proposition 1. The regression functions C p
0
� ( ) and C s0 ( ) of increments have
the following form:
C s V s s
C p p p V p V p p p
0
2
0
1
4
1( ) ( )( ),
( ) ( )
� � � �
� � � ��
� � � � � � �
�
� � � ��V p( ).�
(13)
Here:
V V V V V V� � � � � � � �� � � � � � � �, , : / , , ( , )� � � � � � �� 0 1 .
The Wright–Fisher normalizing functions in (9) have the following representation:
W p W V p( ) ( , ) [ ]� � �� � � �� � � 2 ,
W s W V s( ) ( ) ( )� � �� �
1
4
2 , (14)
W W V V V( ) ( , ) [ / ]� � �� � �� � � �1 .
The regression functions (12), (13) can be calculated as follows:
C s C p C p C p W p p W p p s0 0 0 0
1( ) ( ) ( ), ( ) ( ) ( ), : ( ) /� � � � � �� � �
� � � 2 . (15)
As seen in (13), the regression function with Wright–Fisher normalization in
the transformed form contains a cubic parabola with three real roots:
s s� � � �1 0, �.
Hence there are equilibrium values �� and �:
C C
0 00 0�
� � �( ) ; ( )� � . (16)
The defined in (10) regression functions of increments satisfy the balance
condition
C p C p
0 0 0� �� �( ) ( ) .
STEADY STATE REGIME
Define the fluctuations
� ( ) : ( )S k S kN N� � � . (17)
In the sequel we shall study fluctuations (1)–(3) as primary goal.
The specifics of binary SE allows to calculate the conditional variance:
D S k S k s B s N kN N[ ( ) | ( ) ] ( ) / ,� � � 1 0, (18)
where
B s C s( ) ( )� �1 2 . (19)
The convergence of conditional variance (18) to zero as N � � ensures the
availability of steady state.
Theorem 1. If the initial convergence holds (with probability 1)
S NN
P
( ) ,0
1� �� � �� , (20)
then the following convergence (with probability 1) takes place:
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S k NN
P
( ) ,
1� �� � �� , (21)
for each finite k 0.
Proof of Theorem 1. Just as in [4], we introduce martingale as a sum of
martingale differences:
� N N
k
n
Nn S k C S k( ) : [ ( ) ( ( ))]� � �
�
0
1 . (22)
The quadratic characteristic of martingale (22), taking into account (18), is given
as follows:
� � �
�
� N n
k
n
N
N
B S k: ( ( ))
1
0
. (23)
The boundness of variances ensures convergence (with probability 1) of quadratic
characteristic (23)
� � � �� � � � N n
P
N n
1
0 0, , . (24)
Hence we have the convergence (with probability 1) of martingales (22) (see [5]):
� N
P
n N( ) ,
1
0� �� � � , (25)
for each finite n 0 .
In particular when n � 0 , one has:
M S C S S S C S W SN N N N N N N( ) ( ) ( ( )) � ( ) [ � ( ) ( ( )) / (0 1 0 1 0 00� � � � � ( ))].0 (26)
By assumption (20) of Theorem 1 and properties (13), (14), the term in square
bracket of (26) tends to zero as n � � . Hence the convergence (with probability 1)
� ( )S nN
P
1 0
1� �� � �as
takes place.
By induction, we deduce that for every finite k 1 the convergence (21) takes
place. Theorem 1 is proved.
STOCHASTIC APPROXIMATION
The regression function C s0 ( ) (see (13)) has a multiplier ( )s� � , so there is
the possibility of approximating the SE (1)–(8) by normal process of autoregression
with discrete time k 0, by N � � . Another approach see in [6].
Theorem 2. Under the conditions of Theorem 1 there takes place the limit by
distribution:
N S k C S k k NN N
d
[ ( ) ( ( ))] ( ),� � � �� � � �1 1�� , (27)
for each finite k 0.
Here � 2 is quadratic variation which is expressed through the equilibrium value:
� �2 21� � (28)
and �( ),k k 1, is a sequence of independent, normally distributed random
variables with parameters ( , )0 1 :
E k D k k� �( ) , ( ) ,� � 0 1 1.
Proposition 2 (Approximation). The limit relation (27) forms the basis to
consider the normal process of autoregression
~
( )S kN :
~
( ) (
~
( )) ( ),S k C S k
N
k kN N� � � � 1 1 0
�
� , (29)
as an approximation of the original S kN ( ) with the same nonlinear regression
function
84 ISSN 0023-1274. Êèáåðíåòèêà è ñèñòåìíûé àíàëèç, 2014, òîì 50, ¹ 6
C s s V s s W s( ) ( )( ) / ( )� � � �
1
4
1 2 � , (30)
W s W V s
W V V V V V V
( ) ( ) ( ) ,
( ) / , ( ) / .
� � �
� � � �� � � �
� �
� �
1
4
1
2
(31)
Remark 1. The process of normal autoregression
~
( ),S k kN 0, of course, differs
from the sequence of statistical experiments S k k
N
( ), 0.
In particular, the martingale differences N S k C S kN N[ ( ) ( ( ))]� �1 , generated
by SE, are limited almost surely. However, the fluctuations of normal autoregression
(29) are defined by normal random variables �( ) / ,k N k� 1 0 , and bounded only
in probability. In accordance with Chebyshev inequality
P
N
k C
C N
N
1
1 0 0
2
| ( ) | ,�
�
� �
�
�
�
�
�
�
�
� � ,
for every finite C � 0. So fluctuations in (29) are bounded with the probability,
arbitrarily close to 1 for sufficiently large sample size N .
Remark 2. The normal process of autoregression (28)–(31) retains the property of
persistent regression (7):
E S k S k s C s sN N[
~
( ) |
~
( ) ] ( ), | |� � � �1 1, (32)
with the same regression function (30) as for initial SE.
Proposition 3. In the neighborhood of equilibrium point �, the nonlinear
regression function C s( ) can be approximated by linear regression function
~
( ) (
~
( )) ( )S k C S k
N
kN N� � � �1 1�
�
� ,
C s bs b b s b V W� � � � � �( ) ( ) ( ) , / ( )� � � � � � � �1 1
1
4
2 , (33)
� � � �2 2 21 1 1 4� � � � � �� �, ( ) / /W V V V V .
Proof of Theorem 2. Since the regression function (30) has in denominator
the function W s( ), it greatly complicates the asymptotic analysis of SE. We need the
following lemma.
Lemma 1. The function W p q( , ) (see (9)) admits uniform estimation
0 1� � � �� �max( , ) ( , ) ( )V V W p q W �
for all values of the parameters V V� �, which satisfy the conditions:
0 1� �� �max ( , )V V .
Introduce a martingale as sum of martingale-differences
� N
k
n
N Nn N S k C S k n( ) : [ ( ) ( ( )],� � �
�
0
1 0 . (34)
In order to transform the martingale (34), taking into account the regression
function representation (30), let us introduce the normalized fluctuations
� �N Nk N S k k( ) : [ ( ) ],� � 0 . (35)
Lemma 2. The martingale (34) has the following asymptotic representation:
� � � �N
k
n
N N N
k
n
N Nn k b k
N
k R S k( ) [ ( ) ( )] ( ) ( � ( ))� � � �
� �
0 0
1
1
, n 0 , (36)
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where � ( )S kN is defined in (17),
b V W W V V V V� � � � � �� �1
1
4
1 1 42 2� � � �/ ( ), ( ) / / . (37)
The residual function in (36) has the follwing form:
R s
V
W s
V
W s WN (� ) ( ) � / ( ) ( ),� � � ��
�
�
�
�
�
�
�
!
"
# �
4
2 1
2
2 2� � � � � 1 2� � . (38)
The function R S k kN N( ( )), 0, is bounded in probability under the conditions
of Lemma 1.
Now let us prove, in the same way as in [4], the normal approximation of martingale (34)
or, equivalently, (36)–(38). First, the quadratic characteristics of martingale (34) are the
following:
� � � � �
�
� N n N
k
n
B S k B s C s( ( )), ( ) ( )
0
21 . (39)
Then, by Theorem 1, we find the limit (with probability 1):
� � � �� � � � � �� � � �N n
P
n N
1 2 2 21 1( ) , , , (40)
Further, according to central limit theorem, the main part of martingale (36) converges
in distribution to the sum of normally distributed random variables:
� � � � �N
k
n
N N
d
k
n
n k b k k N0
0 0
1 1( ) : [ ( ) ( )] ( ),� � � � �� � � �
� �
. (41)
At the same time the convergence (41) means that the random values
{ }� � ( ),k k N0 � � are asymptotically independent and have equal variances:
� �� � �� ( ) ( ) ,k k� � �2 21 . (42)
The convergence of martingales (34) means that we have the convergence in
probability (27). Theorem 2 is proved.
CONCLUSIONS
The models of SE given in Proposition 2 and Proposition 3, can be used in
statistical analysis of real experiments.
For this purpose we intend to use the methods of mathematical statistics for
parameters estimation of the normal process of autoregression, namely for estimation
of guiding parameters V �, equilibrium value � and square variance � 2 .
The normal processes of autoregression (29) and (32) can be used to predict the
behavior of real SE, or to detect deviations in the behavior of real statistical data, due
to external conditions change.
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