A MATHEMATICAL MODEL USING FUZZY PEARSON …

A MATHEMATICAL MODEL USING FUZZY PEARSON’S

DISTRIBUTION FOR THE EFFECT OF LEPTIN

aA. Venkatesh* and bK. Vetrivel

aAssistant Professor of Mathematics, A. V. V. M. Sri Pushpam College, Poondi, Thanjavur (Dt), Tamilnadu.

bResearch Scholar, PG and Research Department of Mathematics, A. V. V. M. Sri PushpamCollege, Poondi, Thanjavur (Dt), Tamilnadu..

Abstract

This paper concentrates on the effect of leptin therapy on body weight gain by using

Pearson’s I, III and V distributions. Expected fuzzy mean and variance of the body weight gain

in the given time interval is calculated and the result concludes that the body weight is reduced

due to leptin therapy.

Keywords

Fuzzy person’s distribution, beta distribution, gamma distribution, Leptin

1. Introduction

The groundbreaking discovery of leptin in 1994 ignited the field of obesity research by

providing the first direct evidence for a hormonal system primarily involved in body weight

regulation. Leptin, the protein product of the ob gene [1], is produced only in adipose tissue [2],

[3]. In rodents, leptin administration reduces body weight and adipose tissue mass. In 1994,

leptin, a hormone primarily produced in fat cells, was described for the first time by Zhang and

colleagues [4]. A lack of production of leptin in its active form and a defect in the leptin receptor

are the causes behind the ob/obmouse and the db/dbmouse, respectively [5]. In humans

mutations causing a lack of production of functional leptin or a defect in the leptin receptor have

been shown to cause severe obesity [6], [7].

Leptin is a 16 kDa adipose-derived protein comprising 167 amino acids. The hormone is

coded for by the ob gene which is located on chromosome 7 in humans. The majority of the

leptin hormone is produced in white fat deposits in the body. Levels of leptin are directly

associated with total amounts of fat in the body. Leptin may also be produced in the brown

adipose tissue and by other body parts such as the ovaries, skeletal muscle, fundic glands in the

stomach, placenta, bone marrow, pituitary gland and liver. At the Jackson Laboratory in the

1950s, a certain genetic strain of mice homozygous for the ob (obese) mutation were found to eat

excessive amounts and to be massively obese. Studies of mice with a similar phenotype in the

1960's allowed researchers to characterize the defective gene. During the 1990's, the ob gene was

mapped and eventually identified as responsible for the production of a hormone that could

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 14, Number 4, 2019 (Special Issue) © Research India Publications. http://www.ripublication.com

Page 27 of 34

regulate food intake and body weight. Leptin originates from the Greek word "lepto" meaning

"thin," referring to the fact that the normal allele of the ob gene keeps you thin.

Present paper has relied on secondary data that is a second hand data collected from

various biological and medical papers concentrating on leptin and body weight. Data pertaining

to body weight of rat, infusion of leptin at some time interval have been taken to test whether

hormone leptin is capable of reducing the body weight of the rat. The collected secondary data

were fitted to Pearson-I, Pearson-III and Pearson-V distributions. Statistical tools like fuzzy

mean and fuzzy variance were calculated to prove the statement.

2. Fuzzy Pearson’s Distribution Model

The Pearson distribution is a family of continuous probability distributions. It was first

published by Karl Pearson in 1895 [9] and subsequently extended by him in 1901 and 1916 [10],

[11] in a series of articles on biostatistics. In his original paper, Pearson (1895) identified four

types of distributions (numbered I through IV) in addition to the normal distribution (which was

originally known as type V). The classification depended on whether the distributions

were supported on a bounded interval, on a half-line, or on the whole real line; and whether they

were potentially skewed or necessarily symmetric. A second paper (Pearson 1901) fixed two

omissions: it redefined the type V distribution (originally just the normal distribution, but now

the inverse-gamma distribution) and introduced the type VI distribution. Together the first two

papers cover the five main types of the Pearson system (I, III, IV, V, and VI). In a third paper,

Pearson (1916) introduced further special cases and subtypes (VII through XII).

2.1. Pearson’s Type I Distribution (Beta Distribution)

The probability density function is:

1 1

10

( ) ( ) ( ): , , , ; , ,

( )x a b x

f x a b a x bb a

The mean and variance of Pearson’s Type I distribution are as follows:

)( abaXE

)1()(

)(2

2

abXVar

The alpha cut of the fuzzy mean is

XEXEXE ul ,

where

,},)(

{

,},)(

{

abaSupXE

abaInfXE

u

l


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https://en.wikipedia.org/wiki/Continuous_probability_distribution

https://en.wikipedia.org/wiki/Probability_distribution

https://en.wikipedia.org/wiki/Karl_Pearson

https://en.wikipedia.org/wiki/Normal_distribution

https://en.wikipedia.org/wiki/Support_(mathematics)

https://en.wikipedia.org/wiki/Real_line

https://en.wikipedia.org/wiki/Normal_distribution

https://en.wikipedia.org/wiki/Inverse-gamma_distribution

The alpha cut of the fuzzy variance is

XVXVXV ul ,

where

,},)1()(

)({

,},)1()(

)({

2

2

2

2

abpXuV

abInfXV l

2.2. Pearson’s Type III Distribution (Gamma Distribution)

The probability density function is :

1 xx ef x

The mean and variance of Pearson’s Type III distribution are as follows:

0,

XE

0,2

XVar


XEXEXE ul ,

where

,},{

,},{

SupXE

InfXE

u

l


XVXVXV ul ,

where

,},{

,},{

2

2

SupXV

InfXV

u

l


Page 29 of 34

2.3. Pearson’s Type V Distribution ( Inverse Gamma Distribution)

The probability density function is :

1.

xef x

x

The mean and variance of Pearson’s Type V distribution are as follows:

1,1

XE

2,)2)(1(

2

XVar


XEXEXE ul ,

where

,},,1

{

,},,1

{

SupXE

InfXE

u

l


XVXVXV ul ,

where

,},)2)(1(

{

,},)2)(1(

{

2

2

SupXV

InfXV

u

l

3. Application

Let us take an experiment conducted by Michael G. Dube and others [8], in adult male

Sprague Dawley rats that weighed 225-250gms. The effect of leptin therapy on body weight in

rats over 10 weeks was conducted. The control rats in the experiment shows suppressed induced

weight gain significantly by 24-h kilocalorie consumption weekly and thereby reducing the body

weight significantly.


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Fig 3.1 Rat Body Weight over 10 weeks after Leptin Administration

4. Results

In Pearson’s Type I Distribution, α and β are the shape parameters, a and b are the end

points. The shape parameters and end points of Pearson’s Type I distribution for Fig. 3.1 are

α =0.62206, β=0.59196, a=97 and b=108.

The corresponding triangular fuzzy numbers are

64412.0,62206.0,6.0

68392.0,59196.0,5.0

and the corresponding α cuts are

02206.064412.0,02206.06.0

09196.068312.0,09196.05.0

In Pearson’s Type III Distribution, µ and ð are the shape parameters. The shape

parameters of Pearson’s Type III distribution for Fig. 3.1 are

µ=771.48 and ð=0.13304


96.772,48.771,77

13608.0,13304.0,13.0


00304.013608.0,48.177

00304.013608.0,00304.013.0

In Pearson’s Type V Distribution, λ and α are the shape parameters. The shape

parameters of Pearson’s Type V distribution for Fig. 3.1 are

=844.91 and α=86616.0


82.849,91.844,840

86632,86616,86600

80859095

100105110115120

0 1 2 3 4 5 6 7 8 9 10

% o

f in

itia

l bo

dy

we

igh

t

weeks

rAAV-LEP+HFD(8)


Page 31 of 34


1686632,1686600

91.482.849,91.4840

Table 4.1 Fuzzy mean and variance of Pearson’s Type I Distribution for the body weight gain by

the administration of Leptin

α Values

Fuzzy Mean Values

Fuzzy Variance Values

XE l XuE XV l XuV

0 103 102.3384 14.28571 12.98701

0.1 102.9603 102.3662 14.22663 13.0545

0.2 102.9214 102.3945 14.16638 13.12203

0.3 102.8832 102.4233 14.10507 13.18956

0.4 102.8459 102.4526 14.04282 13.25707

0.5 102.8092 102.4825 13.97971 13.32451

0.6 102.7733 102.5128 13.91585 13.39183

0.7 102.7381 102.5438 13.85133 13.45899

0.8 102.7035 102.5753 13.78622 13.52594

0.9 102.6696 102.6074 13.72059 13.59263

1 102.6364 102.6401 13.65453 13.659

Table 4.2 Fuzzy mean and variance of Pearson’s Type III Distribution for the body weight gain

by the administration of Leptin

α Values Fuzzy Mean Values


XE l XuE XV l XuV

0 592.3077 5680.188 4556.213 41741.54

0.1 592.0616 5691.816 4543.695 41920.63

0.2 591.8167 5703.496 4531.244 42100.92

0.3 591.573 5715.228 4518.86 42282.41

0.4 591.3303 5727.014 4506.541 42465.1

0.5 591.0888 5738.853 4494.288 42649.02

0.6 590.8484 5750.745 4482.1 42834.17

0.7 590.6091 5762.691 4469.977 43020.57

0.8 590.3709 5774.692 4457.917 43208.22

0.9 590.1338 5786.747 4445.921 43397.13

1 589.8978 5798.857 4433.988 43587.32


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Table 4.3 Fuzzy mean and variance of Pearson’s Type V Distribution for the body weight gain

by the administration of Leptin

α Values

Fuzzy Mean Values


XE l XuE XV l XuV

0 103.2181 102.0617 10666.6932 10428.8740

0.1 103.1597 102.1189 10654.6057 10440.5711

0.2 103.1013 102.1761 10642.5391 10452.2883

0.3 103.0429 102.2334 10630.4935 10464.0257

0.4 102.9847 102.2908 10618.4688 10475.7834

0.5 102.9265 102.3483 10606.4649 10487.5612

0.6 102.8684 102.4058 10594.4818 10499.3594

0.7 102.8103 102.4634 10582.5194 10511.1780

0.8 102.7523 102.5210 10570.5777 10523.0169

0.9 102.6944 102.5788 10558.6567 10534.8763

1 102.6365 102.6365 10546.7562 10546.7562

5. Conclusion

In this paper the fuzzy mean and variance for the body weight gain after Leptin treatment

is increased in the lower α cuts and they are decreased in the upper α cuts by using Pearson’s I,

III and V distributions. The major finding of this paper is that rapid weight gain due to intake can

be prevented by Leptin gene therapy and it is viable therapeutic strategy to control weight and

provide protection from adiposity related metabolic disorders.

References

[1]. Zhang Y, Proenca R, Maffei M, Barone M, Leopold L, Friedman JM. 1994, “ Positional

cloning of the mouse obese gene and its human homologue”, Nature, 372, 425-32.

[2]. Masuzaki H, Ogawa Y, Isse N, et al., 1995, “Human obese gene expression. Adipocyte

specific expression and regional differences in the adipose tissue”. Diabetes, 44, 855-858.

[3]. Lonnqvist F, Amer P. Nordfors L, Schalling M., 1995, “Over expression of the obese

(ob) gene in adipose tissue of human obese subjects’, Nat. Med., 1, 950-953.

[4]. Zhang Y, Proenca R, Maffaei M, Barone M, Leopold L, Friedmann JM, 1994, “

Positional cloning of the mouse obese geneand its human homologue”, Nature. 372, 425–

432.

[5]. Lee GH, Proenca R, Montez JM, et al., 1996, “Abnormal splicing of the leptin receptor in

diabetic mice”, Nature, 379, 632–635.


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[6]. Montague CT, Farooqi IS, Whitehead JP, et al. , 1997, “Congenital leptin deficiency is

associated with severe early-onset obesityin humans’, Nature., 387, 903 – 908.

[7]. Cle´ment K, Vaisse C, Lahlou N, et al., 1998, “A mutation in the human leptin receptor

gene causes obesity and pituitary dysfunction”, Nature, 392, 398–401.

[8]. Michael G. Dube, Elena Beretta, HarveenDhillon, Naohiko Ueno, PushpaS.Kalra and

Satya P. Karla,2002, “Central Leptin Gene Therapy Blocks High-Fat Diet-Induced

Weight Gain, Hyperleptinemia and Hyperinsulinemia”, Diabetes, vol51.

[9]. Karl Pearson, 1895, “ Contributions to the Mathematical Theory of Evolution. II. Skew

Variation in Homogeneous Material” Philosophical Transactions of the Royal Society of

London. A, Vol. 186, 343–414. .

[10]. Karl Pearson, 1901, "Mathematical contributions to the theory of evolution, X:

Supplement to a memoir on skew variation". Philosophical Transactions of the Royal

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[11]. Karl Pearson, 1916, "Mathematical contributions to the theory of evolution, XIX: Second

supplement to a memoir on skew variation". Philosophical Transactions of the Royal

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A MATHEMATICAL MODEL USING FUZZY PEARSON …