Ghrelin mathematical modeling and beyond (The big glucose model: the quest for unification)

The big glucose model: the quest for unification
Ghrelin mathematical
modeling and beyond
Jorge Guerra Pires
jorge.guerrapires@graduate.univaq.it
Working Group: Alessandro Borri, Andrea De Gaetano, Pasquale Palumbo,
Costanzo Manes
CNR-IASI - Laboratorio di Biomatematica
UCSC – Largo A. Gemelli 8, 00168, Roma, Italy
Ph: +39 06 30155389 Fax: +39 06 3057845
http://www.biomatematica.it

Overall
• Players for the big glucose model, what we left off last
time.
• Ghrelin mathematical model: an interim modeling theory.
• Ghrelin-leptin: gathering the Jigsaw Puzzle
• On-going
Try to spare some time for “Simulation of insulin regimen and
glucose profiles in Type 1 Diabetic Patient”

The big glucose model: players

The coupling of hormonal responses to nutrient availability is fundamental for
metabolic control(1)
, and our body is a quite nice example when working properly
regarding control engineering.
Metabolism is the important step in which living systems balance the energy
available and the energy demanded, on such a way that the organism will not
find itself in a situation of lacking energy after an abundance(1)
, and organs such
as the brain will have an almost-constant energy supply, namely, glucose.
A literature analysis shows a considerable number of hormones and molecules
involved in the complex process of eating and managing energy; "the regulation of
food intake in humans is an extraordinarily complicated process that researchers
have only begun to understand"(7)
.

Food is equal energy, energy is equal work. We do work from simple tasks such
as sleeping to more complex ones and elaborated tasks such as playing our
favorite sport game.
A number of hormones contribute to rescuing the organism from hypoglycemia
(adrenalin, glucagon, growth hormone, cortisol).

Food Intake vs. Metabolism: body weight control
Schematic viewpoint of the hormones involved in the appetite control and metabolism. In red are important players, but are
not hormones. The dashed arrows intend to show hormones that plays more then one control function on the diagram, the
challenge is to build a complete diagram on this style, even in layers, that is, one hormone may control others such in gene
expression, transcription networks; e.g. some researches show that leptin can control insulin(11). This diagram is
mathematical modeling biased. The "fat" arrow between body weight and metabolism intends to say that the metabolism is
a short-term dynamic process, from seconds to hours. whereas the body weight changes in a long-term scale, from days to
months. Source: own elaboration.

Insulin
Insulin is a peptide hormone produced by beta cells in the pancreas.
It regulates the metabolism of carbohydrates and fats by promoting the
absorption of glucose from the blood to skeletal muscles and fat tissue and by
causing fat to be stored rather than used for energy. Insulin also inhibits the
production of glucose by the liver.
Wikipedia contributors, "Insulin," Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/w/index.php?title=Insulin&oldid=662494754
(accessed May 20, 2015).
Insulin, essentially, lowers blood glucose levels as it increases glucose uptake in
peripheral tissues(7)
and liver.
Insulin levels in the blood reflect both circulating energy (i.e., glucose) and stored
energy (i.e., adipose tissue) (5,7)
.

Insulin
In fact, it is claimed to reflect stored energy in visceral fat tissues (i.e. around organs),
whereas leptin in subcutaneous fat tissues (i.e. under skin).
Circulating insulin levels constitute a dynamic metabolic switch, signaling the fed
state and nutrient storage (anabolic pathways) when elevated, or starvation and
nutrient mobilization (catabolic pathways when decreased(5,1)
.

Glucagon
Glucagon is a peptide hormone, produced by alpha cells of the pancreas, that
raises the concentration of glucose in the bloodstream. Its effect is opposite
that of insulin, which lowers the glucose concentration. Glucagon generally
elevates the concentration of glucose in the blood by promoting
gluconeogenesis and glycogenolysis.
Wikipedia contributors, "Glucagon," Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/w/index.php?title=Glucagon&oldid=662846755
(accessed May 20, 2015).
Glucagon is a hormone secreted by the alpha cells of the pancreas that decreases
food intake(7)
.
Glucagon is the “hormone of starvation,” in contrast to leptin known as the
"the hormone of satiety", and levels of glucagon do increase when someone
fasts.

Glucagon
Glucagon mainly stimulates glucose production either by breaking down glycogen
(i.e. Glycogenolysis) or by producing more glucose (i.e. gluconeogenesis) in the
liver, particularly during the fasting condition or when the body has an increased
need for glucose(7)
.
Glucagon seems to work directly on the liver to inhibit food intake (7)
.

Leptin
Leptin is a signal of adiposity; it triggers food intake when it is low and inhibit
food intake when it is high. It has been proposed as a co-worker with insulin to
control and monitor body adiposity and metabolism(10)
.
It is not clear yet, but it seems to work with ghrelin on the food intake control. It
is a long-term hormone signal, whereas ghrelin and insulin are short-term
hormonal signals.

Glycogen
Glycogen is one of the molecules by which energy is stored in humans, the
second is in fat cells as triglyceride. It is stored in the liver and released in time of
hypoglycemia, it is triggered by glucagon.

Triglycerides
Triglycerides (fat and oil) consist of three fatty acid molecules joined to a
molecule of glycerol.
Hydrolysis of triglycerides within adipose tissue releases free fatty acids into
the blood. Free fatty acids can be used as an immediate source of energy by
many organs; they can also be converted by the liver into derivatives called
ketone bodies.
Triglycerides can be hydrolyzed into glycerol and fatty acids. The latter are of
particular importance because they can be converted into numerous molecules
of acetyl CoA that can enter Krebs cycles and generate a large amount of ATP.

Ghrelin
Ghrelin is an important hunger signal secreted by the stomach; it was
discovered in 1999, about four years after its sister leptin.
Ghrelin secretion rises between meals[1], when the stomach is empty, and
stimulates hunger. As the stomach fills during a meal, the secretion of ghrelin
rapidly falls and hunger is thereby reduced.
However, one recent study demonstrated raised levels of ghrelin in dieters who
lost weight. If this raised ghrelin level enhances appetite, it may partially explain
why it is so difficult for most dieters to maintain their weight loss. It is produced
by several points within the body such as pancreas ( -cells), stomach, andɛ
heart(13).
[1] It has been some controversies on the literature regarding its secreting laws.

(Poly) Peptide YY
A recently discovered hormone secreted by the small intestine, named
polypeptide YY (PYY), regulates hunger on a more intermediate-time basis.
See scheme in fig.4. Basically it is reported to count glucose absorption, it
inhibits the intake of food; therefore opposing ghrelin and supporting leptin
and insulin.

Cholecystokinin
Another hormone that regulates eating is the intestinal hormone
cholecystokinin (CCK)[1]. Secretion of CCK rises during and immediately
after a meal and suppresses hunger (promotes satiety).
[1] This hormone was the one responsible for the discovery of leptin, it was thought CCK played the role on regulating food
intake now granted to leptin and ghrelin.
CCK thus acts antagonistically to ghrelin, helping to reduce appetite immediately
after a meal. Cholecystokinin is a hormone produced primarily in the duodenum
and jejunum parts of the small intestine, but also in the brain. It decreases meal
size(7).
Cummings and Overduin(8) note that CCK is involved in short-term
satiation, acutely shortening mealtime, rather than long-term control.

Cholecystokinin
The intestinal satiety hormones, including CCK, PYY, glucagon-like peptide-1
(GLP-1) and others, are believed to stimulate sensory neurons of the vagus
nerve, which communicate the satiety signals to the CNS.

Amylin
Amylin is a peptide hormone also secreted after meals, along with insulin, by
the beta cells of the pancreas.
Amylin inhibits the stomach from emptying, as well as inhibiting gastric acid and
glucagon secretion. It has the capacity to reduce food intake and meal size(7).
Amylin is secreted in proportion to food intake and can be considered a
hormone of satiety(7). Amylin does cross the blood-brain barrier and works
directly on certain areas of the brain, and is considered a neuro-endocrine
hormone(7).

Somatostatin
In the pancreas, somatostatin is produced by the delta cells of the islets of
Langerhans, where it serves to block the secretion of both insulin and glucagon
from adjacent cells.
Insulin, glucagon, and somatostatin act in concert to control the flow of nutrients
into and out of the circulation(6).

Glucagon-like peptide (GLP-1)
Glucagon like peptide (GLP-1) is a very powerful stimulator of insulin secretion
from the pancreatic islets.(5)
GLP-1 is released in response to meal intake(9). GLP-1 also appears to
be a physiological regulator of appetite and food intake(9).

Neuropeptide Y
Neuropeptide Y is one of the most prevalent peptides throughout the brain
(including the hypothalamus) and even the sympathetic nervous system.
It stimulates feeding behavior and weight gain(7). Notably, both insulin and leptin,
which decrease food intake, inhibit neuropeptide Y, and fasting increases its levels.

Adiponectin
Adipocytes have been found to secrete regulatory molecules, collectively
termed adipokines, which regulate hunger, metabolism, and insulin
sensitivity(5)
.
One of those adipokines is called adiponectin. Adiponectin, discovered in the
1990s, is an amino acid protein hormone that is secreted in adipose tissue but
found circulating in blood(7).
In contrast to levels of leptin, which are higher in obese people, levels of
adiponectin are significantly lower in the obese(7).
Furthermore, levels are lower in people who have insulin resistance, that is, those
with high levels of insulin and abnormal glucose tolerance test results(7). At least
in animals, administration of adiponectin is known to enhance the action of insulin,
and it has been found to lower circulating levels of glucose(7).

Remarks
•As it can be seen, the functions and roles seem fuzzy, most hormones seem
to be doing the same thing.
•The key challenge from a mathematical modeling standpoint is how to
separate properly the workings of each hormone.
•One possibility is quite simple: like in the net force, we have several players,
but just one appears, an imaginary force, which is coordinated by all of the
hormones.
•If so, the question is how to project each component on a such a way to see
clearly the contribution of each of them.

References (The big glucose model)
1. Alfa RW, Park S, Skelly KR, Poffenberger G, Jain N, Gu X, Kockel L, Wang J, Liu Y, Powers AC, Kim
SK. Suppression of insulin production and secretion by a decretin hormone. Cell Metab. 2015 Feb
3;21(2):323-33. doi: 10.1016/j.cmet.2015.01.006.
2. K. N. Frayn. Metabolic Regulation: A Human Perspective. Third Edition. Wiley-blackwell. 2010.
3. J. Tam, Dai Fukumura, and Rakesh K. Jain. A mathematical model of murine metabolic regulation by
leptin: energy balance and defense of a stable body weight. Cell Metab. 2009 January 7; 9(1): 52–63.
doi:10.1016/j.cmet.2008.11.005.
4. Palumbo P, Ditlevsen S, Bertuzzi A, De Gaetano A, Mathematical modeling of the glucose–insulin
system: A review, Mathematical Biosciences 244 (2013) 69–81.
5. Fox SI. Human Physiology. twelfth edition. McGraw Hill: 2011.
6. Encyclopædia Britannica Online, s. v. "somatostatin", accessed aprile 24, 2015,
http://www.britannica.com/EBchecked/topic/553961/somatostatin.
7. Karasu, SR. Karasu, TB. The Gravity of Weight: a clinical guide to Weight Loss and Maintenance.
American psychiatric Publishing, Inc. 2010.
8. Cummings DE, Overduin J. Gastrointestinal regulation of food intake. Review series. J. Clin. Invest.
117:13–23 (2007).

References (The big glucose model)
9. Holst, JJ. The Physiology of Glucagon-like Peptide 1. American Physiological Society. Physiol Rev 87:
1409–1439, 2007.
10. S. C Benoit, D. J Clegg, Ra. J Seeley, and S. C Woods. Insulin and Leptin as Adiposity Signals. The
Endocrine Society. 2004.
11. Castracane VD, Henson MC, editores. Leptin. Spring: 2006.
12. Gertler A. Leptin and Leptin Antagonists. Landes Bioscience, 2009.
13. Machado Romero CE, Zanesco A. O papel dos hormônios leptina e grelina na gênese da obesidade,
Rev. Nutr. vol.19 no.1 Campinas Jan./Feb. 2006. [citado 2015 Apr. 22]. Disponível em:
http://www.scielo.br/pdf/rn/v19n1/28802.pdf.
14. "Energy Balance" by KmpH - Own work. Licensed under CC BY 3.0 via Wikimedia Commons -
http://commons.wikimedia.org/wiki/File:Energy_Balance.png#/media/File:Energy_Balance.png

References
Magner, LN, A history of medicine, second edition, Taylor&Francis, 2005.
Keith N. Frayn, Metabolic regulation: a human perspective, third edition, Wiley-blackwell, 2010.
Signore, AP; Zhang, F; Weng, Z.; Gao, Y; Chen, J. leptin neuroprotection in the CSN: mechanisms and
therapeutic potentials. International Society for Neurochemistry, J. Neurochem. (2008) 106.
Jeffrey M Friedman, A tale of two hormones, Nature Medicine 16, 1100–1106 (2010)
doi:10.1038/nm1010-1100.
J. Tam, Dai Fukumura, and Rakesh K. Jain. A mathematical model of murine metabolic regulation by
leptin: energy balance and defense of a stable body weight. Cell Metab. 2009 January 7; 9(1): 52–63.
doi:10.1016/j.cmet.2008.11.005.

Ghrelin Mathematical
modeling
Note of the author. the discussions herein are
merely a starting point for more advanced models
and insights.
with embedded simulations

Ghrelin Mathematical modeling
Ghrelin, produced mainly in the stomach, is an appetite stimulating hormone; i.e. it
triggers the need for food, our appetite.
It has been discovered in 1999 by Japanese scientist, but largely spread-out by
British groups, and so then it has been a quite important piece for taking in the
workings of feeding patterns and behaviors.
Ghrelin is an amino acid peptide, related to growth hormone, that is secreted
primarily in the stomach but is found throughout the gastrointestinal system and
even in the hypothalamus and amygdala, among other sites, such as the heart
and pancreas. Some claims that the name comes from Growth Hormone
releasing, by shorting and gathering, we encounter ghrelin.
But exactly how ghrelin exerts its effect is not clear, neither how it is produced,
e.g. the complete profile for triggering ghrelin activation and inhibition.
Introduction

The complex process of eating
The complex process of
eating. Source: Nelson
and Cox (2004, p.912).

Ghrelin Dynamics: a qualitative look-upon
The general workings of Ghrelin, connecting the brain with the digestive system, and vise versa. Essentially,
we have the response of the system by eating when ghrelin is high, blue dots. We have an output neuron
control (e.g. amylin), represented by green dots, called hormone y, it is an imaginary hormone. Ghrelin falls in
the bloodstream to achieve the brain (i.e. hypothalamus, the arcuate nucleus). The brain respond by reducing
the appetite, which is physically seen by eating less. The stomach is under two physical forces, namely,
stretching force due to food incomings, and restoring forces, due to the plasticity of the walls of the stomach.
Source: own elaboration.

Force Balance
Force balance for the stomach in the eating dynamics. The stomach is under two
forces, one stretching and other storing its resting shape. The stretching is result
of food entering the stomach, whereas the restoring is the result of the stomach
trying to stay on its minimal size configuration. Simply thinking, the stomach is a
two input container. Source: own elaboration.

Stretching force

Stretching force
Transforming a three-dimensional oscillation into an one dimensional.
Source: own elaboration.

Stretching force
Different food densities, the blue line is for a food three times denser than
the red.
Source: own elaboration, after the model is ready.

Stretching force
Different food densities, changing within the meal time.
Source: own elaboration, after the model is ready.

Restoring force

Resultant (net) force

Mass Balance for the stomach
outin
food
rr
dt
dm
−=
n
in
n
brain
n
brain
inin
kGh
Gh
r
+
= α
stomach
out
out m
ky
y
r 





+
−= σξ

Mass Balance for the stomach: rates in

Mass Balance for the stomach rates in

Mass Balance for the stomach: general ghrelin profile
Ghrelin seems to trigger hunger in a complex way, in general, just increase of
ghrelin will not trigger hunger. Therefore, K must be big.
Secondly, the activation must be in the style of "0 and 1", food is in general
massive, a single swallow will certainly fulfill the stomach in-pipe. Thus, n must
be high.

Mass Balance for the stomach:
simulations

Mass Balance for the stomach: mass
that comes out
stomach
out
out m
ky
y
r 





+
−= σξ 1≥σ
,
Where: y is a hypothetical hormone, it supposes to work as a resultant (i.e.
net) force. Herein we take amylin as being this force, but we would have to
take into account others such as gastrin, or even cholecystokinin, for being
more precise.

Mass Balance for the stomach: mass
that comes out

Ghrelin differential equation: differential equations, plasma
( )
( )
brain
plasmaplasmabrainbrainplasmagh
out
outgh
gh
plasma
Ghkappa
GhGhKGhGFRCl
r
kf
f
dt
dGh
*
/log**
1
1
+
−
−







 −
−
+
−=
λ
β
α
αβ
1≤α
,

Ghrelin differential equation: differential equations, brain
( )
brain
plasmaplasmabrainbrain
brain
Ghkappa
GhGhK
dt
dGh
*
/log
−
= λ

Ghrelin differential equation: differential equations, brain
The Gompertz dynamics, in general applied to descript tumor cells growth

Hormone Y dynamics
yGFRClr
t
dy
yiny **−= σ
stomach
out
out m
ky
y
r 





+
−= σξ

Ghrelin-Leptin
Mathematical modeling

Ghrelin-Leptin Mathematical
modeling
Leptin (Murine model)

modeling
Ghrelin (just-seen model)

modeling
A Jigsaw puzzle

modeling
A mechano-hormonal control
system

modeling

modeling
This is problem from the old-leptin model, maybe I know how to fix it.
You should not allow the mode to spend more energy then it has, it so, it
is negative.

Multi-time-scale modeling: strategies
Strategy 0: steady-state analysis
Equal to zero all the slow dynamics, solve the algebraic equations, substitute
the values into the differential equations
Ex. In gene expression, you can assume that the mRNA dynamics is slow
compared to protein
Advantage. Straightforward to understand and implement.
Disadvantage. We lose the slow-dynamics once and for all.

Strategy 1: unit conversions
Change all the time-scale units for the smallest one, e.g. From days to hours,
from week to days.
Ex. 4 ng/week = 0.58 ng/day, 10 km/h= 0.28 m/s
Disadvantage. The parameters can be “distorted”, e.g. what is small in hour
will be even smaller in seconds or minutes, analytically a piece of cake, but
computationally (numerically) it can create problems.

Strategy 2: slow down the rates
Multiply the rates in order to make them equivalent.
Ex. ( )
( )yxg
dt
dy
yxf
dt
dx
,
,
=
=
dt
dy
dt
dx
=α ( )
( )yxg
dt
dy
yxf
dt
dx
,
,
=
=
α
Disadvantage. When slowed down, it might affect the original dynamical
systems in a negative way.

Strategy 3: two-layers of simulations, “parallel universes”
Advantage. Straightforward to understand, but a taught call to implement.
Disadvantage. It might be difficult to now when send information, for instance,
when simulating disorders, it will create “jumps”, even in normal conditions.

Triglycerides and leptin
Fats (lipids, as they are referred to scientifically) consist of three fatty acids
attached to the sugar alcohol glycerol. This structure is called a triglyceride and
thus all fats are triglycerides.
With increased insulin secretion, more storage of triglycerides in fat
cells.

Experiments demonstrate that triglycerides can reduce leptin transport across the
blood–brain barrier (Banks et al. 2004).
Diminished transport of leptin across the blood–brain barrier would occur in the
presence of higher triglyceride levels in obese individuals, and is therefore
thought to be directly responsible for the lack of elevated brain leptin levels in
obesity (Signore et al, 2008).
This change in the leptin transport system may be an adaptive response to
fasting, during which the body begins to break down and release triglycerides
and where any additional anorexigenic leptin signal would be counterproductive
for long-term survival (Signore et al, 2008).

White adipose tissue, or white fat, is where most of the triglycerides in the
body are stored. When fat stored in adipose tissue is going to be used as an
energy source, lipase enzymes hydrolyze (breakdown) triglycerides into
glycerol and free fatty acids in a process called lipolysis.
These molecules (primarily the free fatty acids) serve as blood-borne energy
carriers that can be used by the liver, skeletal muscles, and other organs for
aerobic respiration.

)mbloodstreatheto()mbloodstreathefrom( +−=
dt
dTg
22
2
1
AN
BN
K
N
Nr
dt
dN
b
B
+
−





−=
K
)mbloodstreatheto()mbloodstreathefrom( +−=
dt
dTg
22
2
1
AN
BN
K
N
Nr
dt
dN
b
B
+
−





−=

Ghrelin and insulin
Variations in ghrelin and insulin
relative to meal times.
(a) Plasma levels of ghrelin rise
sharply just before the normal
time for meals (7 a.m.
breakfast, 12 noon lunch, 5:30
p.m. dinner) and drop
precipitously just after meals,
paralleling the subjective
feelings of hunger.
(b) Insulin levels rise immediately
after each meal, in response to
the increase in blood glucose
concentration.
(Nelson and Cox, 2008)

Differential equation for leptin: from plasma to
brain
)mbloodstreatheto()mbloodstreathefrom(
_
−=
dt
braindLep
22
2
1
AN
BN
K
N
Nr
dt
dN
b
B
+
−





−=

References
Banks W. A., Coon A. B., Robinson S. M., Moinuddin A., Shultz J. M., Nakaoke R. and Morley J. E.
(2004) Triglycerides induce leptin resistance at the blood-brain barrier. Diabetes 53, 1253–1260.
Signore, AP; Zhang, F; Weng, Z.; Gao, Y; Chen, J. leptin neuroprotection in the CSN: mechanisms and
therapeutic potentials. International Society for Neurochemistry, J. Neurochem. (2008) 106.
David L. Nelson; Michael M. Cox, Lehninger principles of biochemistry, Fifth Edition, W.H Freeman and
Company, New York, 2008.

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