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

[email protected]

Working Group: Alessandro Borri, Andrea De Gaetano, Pasquale Palumbo, Costanzo Manes

CNR-IASI - Laboratorio di BiomatematicaUCSC – Largo A. Gemelli 8, 00168, Roma, ItalyPh: +39 06 30155389 Fax: +39 06 3057845http://www.biomatematica.it

Run for your lives!

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

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).

Insulin

Glucagon

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.

http://en.wikipedia.org/w/index.php?title=Glucagon&oldid=662846755

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

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

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

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

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 (ɛ-cells), stomach, and heart(13).

[1] It has been some controversies on the literature regarding its secreting laws.

(Poly) Peptide YY

(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 (CCK)

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

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

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)

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

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

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

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

ReferencesMagner, 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, 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


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


outinfood rrdt

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nin

nbrain

nbrain

inin kGhGhr

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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.

Ghrelin Mathematical modelingMass Balance for the stomach: simulations

Ghrelin Mathematical modelingMass Balance for the stomach: mass that comes out

stomachout

out mkyyr

1

,

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.

Ghrelin Mathematical modelingMass Balance for the stomach: mass that comes out

Ghrelin Mathematical modelingGhrelin differential equation: differential equations, plasma

brain

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Ghrelin Mathematical modelingGhrelin differential equation: differential equations, brain

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Ghrelin Mathematical modelingGhrelin differential equation: differential equations, brain

The Gompertz dynamics, in general applied to descript tumor cells growth

Ghrelin Mathematical modelingHormone Y dynamics

yGFRClrtdy

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Ghrelin-Leptin

Mathematical modeling

Ghrelin-Leptin Mathematical modelingLeptin (Murine model)

Ghrelin-Leptin Mathematical modelingGhrelin (just-seen model)

Ghrelin-Leptin Mathematical modeling

A Jigsaw puzzle


A mechano-hormonal control system


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.

On going....

Multi-time-scale modeling: strategiesStrategy 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.

Multi-time-scale modeling: strategiesStrategy 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.

Multi-time-scale modeling: strategiesStrategy 2: slow down the rates

Multiply the rates in order to make them equivalent.

Ex.

yxgdtdy

yxfdtdx

,

,

dtdy

dtdx

yxgdtdy

yxfdtdx

,

,


Disadvantage. When slowed down, it might affect the original dynamical systems in a negative way.

Multi-time-scale modeling: strategiesStrategy 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 leptinFats (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.

Triglycerides and leptin

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 counterproductivefor 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.


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Ghrelin and insulinVariations 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

)mbloodstrea theto()mbloodstrea thefrom(_

dtbraindLep

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2

1AN

BNKNNr

dtdN

bB

ReferencesBanks 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.

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

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