Transport of Fatty Acid Molecules by Lipoprotein Micelles in the Human

Transport of Fatty Acid Molecules by Lipoprotein

Micelles in the Human Body

By:

Minxue Jia

A thesis submitted to Johns Hopkins University in conformity with the requirements for the

degree of Master of Science in Engineering

Baltimore, Maryland

May, 2018

Abstract

Though the death rate from coronary heart disease (CHD) has declined over the last several decades, CHD still remains one of the leading causes of death in the United States.

Multiple lines of evidence have confirmed that Apo B-containing lipoproteins are involved in the development of atherosclerosis. The function of Apo B-containing lipoproteins is to transfer fatty acids and other lipids to peripheral organs.

Two computational and mathematical methods, a deterministic ODE model and the stochastic Gillespie algorithm, are applied to model the complex lipoprotein transport and lipid metabolism of the human body for a better understanding of human’s lipid transfer system and its relationship to atherosclerosis. The ODE model is a set of differential equations which tracks the lipoproteins with an assumption of one-step transition between lipoprotein groups. While the

Gillespie algorithm simulates the possible states for each lipoprotein at each step, which enables subtle track of lipoprotein transition. The two modeling methods provide a way to look at fatty acid transport qualitatively and quantitatively.

The goal of this research is to obtain insight into fatty acid transport in the human body and develop strategies to prevent and treat atherosclerosis.

Thesis Committee:

Marc D. Donohue, Professor of Chemical Engineering

Michael J. Betenbaugh, Professor of Chemical Engineering

Table of Contents Abstract...... ii Acknowledgements ...... iv List of Figure ...... v List of Table ...... vi Chapter 1. Introduction ...... 1 1.1 Atherosclerosis ...... 2 1.2 Apo B Lipoproteins ...... 3 1.3 HDL & Reverse Cholesterol Transfer ...... 9 1.4 Lipase ...... 10 1.5 CETP & PLTP ...... 11 1.6 Small dense LDL ...... 12 Modeling and Results ...... 14 Chapter 2. Ordinary Differential Equation Model ...... 14 2.1 The lipoprotein species and transition ...... 14 2.2 Organ lipids and formation of lipoprotein ...... 15 2.3 Cholesterol ester transfer protein and sdLDL ...... 15 2.4 LDL receptor & LDL receptor related protein ...... 16 2.5 Results ...... 16 Chapter 3. Modified Gillespie algorithm: ...... 28 3.1 Grouped Gillespie algorithm ...... 28 3.2 The three tracks ...... 29 3.3 Organ lipids and formation of lipoprotein ...... 29 3.4 Circulation: ...... 30 3.5 Lipoprotein lipase ...... 30 3.6 Hepatic lipase ...... 31 3.7 Cholesteryl ester transfer protein ...... 31 3.8 LDL receptor ...... 32 3.9 Result ...... 32 3.10 Limitation ...... 34 Chapter 4. Conclusion ...... 36 Reference ...... 37 Appendix ...... 44 Appendix I Parameter for ODE ...... 44 Appendix II Differential Equation & Matlab Code for ODE...... 46 Appendix III Matlab Code for Modified Gillespie algorithms ...... 76 Curriculum Vitae ...... 91

iii

Acknowledgements

I would like to greatly thank Professor Donohue for being my research advisor and for all of his guidance. I learned to model and apply computational & mathematical methods to solve complex problems from Professor Donohue’s teaching and mentoring. His support has always pushed me to strive in his lab.

I would also like to thank Professor Betenbaugh for being on my thesis committee. Not only did his feedback help me understand my own research better, he kept me focused on the task at hand. In addition, I would like to thank the other students Denis Routkevitch, Gabriella

Russo, Victoria Laney, Junyi Rao and Tiankai Zhang, who have helped me for my research work.

Lastly, I would like to thank my friends and family, especially my parents, who have supported and helped me throughout this process and deserve all of the credit.

List of Figure

FIGURE 1. APOLIPOPROTEIN TRANSFER BETWEEN LIPOPROTEINS 6 FIGURE 2. LIPOPROTEIN METABOLISM 8 FIGURE 3. CETP ACTIVITY AND LDLS & HDLS 16 FIGURE 4. CM AND CMR CONCENTRATION AT HOMEOSTASIS STATE 17 FIGURE 5. VLDL AND IDL CONCENTRATION AT HOMEOSTASIS STATE 18 FIGURE 6. LDL CONCENTRATION AT HOMEOSTASIS STATE 18 FIGURE 7. HDL CONCENTRATION AT HOMEOSTASIS STATE 19 FIGURE 8. LIPID CONCENTRATION AND CHOLESTEROL CONTENT IN VLDL AT HOMEOSTASIS STATE 19 FIGURE 9. CM AND CMR CONCENTRATION AFTER INCREASED DIETARY LIPID INTAKE 20 FIGURE 10. VLDL & IDL CONCENTRATION AFTER INCREASED DIETARY LIPID INTAKE 21 FIGURE 11. LDL CONCENTRATION AFTER INCREASED DIETARY LIPID INTAKE 22 FIGURE 12. HDL CONCENTRATION AFTER INCREASED DIETARY LIPID INTAKE 22 FIGURE 13. HEPATIC TRIGLYCERIDE & CHOLESTEROL AFTER INCREASED DIETARY LIPID INTAKE 23 FIGURE 14. LIPID CONCENTRATION AND CHOLESTEROL CONTENT IN VLDL AFTER INCREASED DIETARY LIPID INTAKE 24 FIGURE 15. VLDL & IDL CONCENTRATION WHEN VLDL GENERATION IS LIMITED 25 FIGURE 16.LDL CONCENTRATION WHEN VLDL GENERATION IS LIMITED 25 FIGURE 17. HDL CONCENTRATION WHEN VLDL GENERATION IS LIMITED 26 FIGURE 18. LIPIDS CONCENTRATION AND AVERAGE CHOLESTEROL CONTENT IN VLDL WHEN VLDL GENERATION IS LIMITED 27 FIGURE 19. COMPOSITION OF EACH TYPE OF LIPOPROTEIN 33 FIGURE 20. BODY CONCENTRATION THROUGH TIME 33 FIGURE 21. ORGAN COMPOSITION. 34

List of Table

TABLE 1. LIPOPROTEIN SIZE, DENSITY, CONCENTRATION, COMPOSITION AND

APOLIPOPROTEI 4

TABLE 2. VOLUME AND BLOOD FLOW RATE OF ORGANS 44

TABLE 3. COMPOSITION OF LIPOPROTEINS IN MOLE 44

TABLE 4. TRIGLYCERIDE CONCENTRATION OF ORGANS 44

TABLE 5. CHOLESTEROL CONCENTRATION OF ORGANS 45

TABLE 6. FATTY ACID PROFILE AND TRIGLYCERIDE MOLAR MASS 45

Chapter 1. Introduction

The death rate from coronary heart disease (CHD) has declined noticeably recently, but

CHD remains one of the leading causes of death in the United States. 1 The CHD is a disease in which a waxy substance called plaque builds up inside the coronary arteries. Coronary arteries are the heart’s network of blood vessels, and they supply oxygen-rich blood to the body. The lack of oxygen-rich blood to portions of the heart muscle leads to ischemia of myocardial tissues and consequent alteration of heart function. 2 Atherosclerosis gives rise to CHD through slowly progressing lesion formation and luminal narrowing of arteries of the heart that supply the blood for maintaining normal cardiac function.3 The cause of atherosclerosis appears to be lipid retention, oxidation, and modification, which provokes chronic inflammation at susceptible sites in the major conduit arteries, ultimately causing thrombosis or stenosis. 4

Epidemiology has shown that elevated levels of cholesterol play a critical role in the development of atherosclerotic disease. 5 Multiple lines of evidence have confirmed that cholesterol-rich LDL and other a-apolipoproteins (ApoB)-containing lipoprotein, including very low-density lipoprotein (VLDL) and intermediate density lipoprotein (IDL) are directly implicated in the development of atherosclerotic disease.6 This project is an attempt to better understand CHD, lipoprotein transport and lipid metabolism using a computational scheme.

1.1 Atherosclerosis

Atherosclerotic lesions, as Atherosclerotic plaques, typically present as asymmetric focal thickenings of the innermost layer of the artery, the intima. The plaques are composed of cells, connective tissue elements, lipids, and debris. The cells in the plaque are primarily inflammatory or immune cells, and the remainder are vascular endothelial and smooth muscle cells. The plaque formation involves several steps.

The atherosclerosis process is initiated, when lipoproteins infiltrate the artery wall and accumulate in the extracellular matrix.7 The Low-density-lipoproteins undergo enzymatic attack or non-enzymatic oxidation in the intima, result in a release of bioactive phospholipids which can activate the endothelial cell. Endothelial cell’s activation causes circulating monocytes and other inflammatory cells to adhere to endothelial cells and migrate into the intima. The cells in the intima produce macrophage-colony stimulating factor, which induces monocytes to differentiate into macrophages.

Macrophages express a variety of scavenger receptors, which can interact with lipoproteins, including SR-A1, CD36, and lectin-like oxLDL receptor-1. Also, macrophages contain several cholesterol transporters that are involved in reverse cholesterol transport. The lipids or lipoprotein would be internalized by macrophage to release cholesterol. The acetyl-CoA acetyl-transferase (ACAT1), an enzyme in macrophages, transforms free cholesterol to cholesteryl ester. Another enzyme in macrophages, neutral cholesterol ester hydrolase (NCEH), converts cholesterol ester to free cholesterol that can exit the macrophages via cholesterol transporters.8 In atherosclerosis, the scavenger receptor expression of macrophage is upregulated and allows excessive oxLDL uptake, while the cholesterol transporter expression is

2 downregulated. The combination results in deposition of free cholesterol and cholesteryl ester in macrophage and their conversion to foam cells.

1.2 Apo B Lipoproteins

The term ‘cholesterol’, ‘LDL’, and ‘LD cholesterol (LDL-C)’are frequently conflated or used interchangeably, which causes considerable confusion. Cholesterol is an essential component of the mammalian cell membrane and also is a precursor for bile acid and steroid hormones. Cholesterol and other lipids are transported to peripheral cells by ApoB -containing micelles in plasma, and the classification of VLDL, IDL & LDL depends on particles size of micelles. LDL particles, the smallest ApoB -containing micelle, constitute around 93 percent of circulating ApoB-containing micelles in fasting blood. (Table 1) In clinical practice, the plasma

LDL level is estimated from the cholesterol concentration of plasma LDL (LDL-C) instead of measuring LDL concentration directly.

Triglycerides are the most energy-dense molecules in the human body and serve as a primary source of fuel, but triglycerides are insoluble in blood. To be transported from the intestine and the liver to other organs of the body, triglyceride should be packaged in a form compatible with the aqueous environment in plasma. The lipoproteins are complex particles with a central core containing cholesterol ester and triglyceride surrounded by free cholesterol, phospholipids and Apo-lipoproteins, which regulate lipoprotein metabolism. Plasma lipoproteins are classified into six groups based on size, lipid composition, and the Apo- lipoproteins on their surface: chylomicrons, chylomicron remnants, very low density lipoprotein

(VLDL), intermediate density lipoprotein (IDL), low density lipoprotein (LDL) and High

Density Lipoprotein (HDL).10 Table 1 below shows the composition and Apo-lipoproteins of each type of lipoprotein micelle.

Chylomicron VLDL IDL LDL HDL

Density (g/ml) 0.95 0.978 1.0125 1.041 1.1365

Diameter(cm) 5.00E-05 5.50E-06 3.00E-06 2.15E-06 8.50E-07

Concentration 12.5 69 32 1272 35000 (nmol/L)

Protein (g / per) 1.20E-15 6.80E-18 2.10E-18 1.20E-18 1.70E-19

Triacylglycerol 5.30E-14 4.70E-17 4.40E-18 3.20E-19 1.50E-20 (g / per)

Free Cholesterol 1.20E-15 6.00E-18 1.00E-18 4.30E-19 1.50E-20 (g / per)

Cholesterol ester 1.90E-15 1.00E-17 3.30E-18 2.30E-18 5.80E-20 (g / per)

Phospholipids 4.40E-15 1.50E-17 3.10E-18 1.20E-18 1.00E-19 (g / per) A-1, A-2, A-4, B-100, C- A-1, A-2, Apo-lipoprotein B-100, C-1, B-48, C-1, C-2, 1, C-2, C- B-100, E C-1, C-2, Composition C-2, C-3, E C-3, E 3, E C-3, D, E Table 1. Lipoprotein Size, Density, Concentration, Composition and Apolipoproteins 9 11

The Apo-lipoproteins on the surface of the micelles bind to receptors and serve as structural role. They also can act as enzyme cofactor and cell surface ligand to regulate lipoprotein metabolism. As an untransferable Apolipoprotein, Apo B is the main structural surface protein found on chylomicrons, chylomicron remnants, VLDL, IDL and LDL. There are two forms of Apo B, Apo B 100 and Apo B 48. Both forms are products of the same structural gene on chromosome 2. 12 Apo B 48 is produced in the proximal small intestine and secreted

4 within chylomicrons. Apo B 100 is synthesized in the liver and secreted with VLDL, which subsequently is metabolized to form LDL in plasma. Apo B containing the LDL receptor binding domain, interacts with LDL receptor for endocytosis. Therefore, VLDL, IDL and LDL can be removed from plasma by binding to LDL receptors.

Since chylomicrons are secreted in the intestine, dietary lipid is packaged by Apo B 48.

Chylomicrons enter the blood circulation through the lymphatic system at the left subclavian vein.13 The chylomicrons also contains some Apo AI, which is synthesized in intestine. The

ApoAI are transferred spontaneously to the HDL as soon as the chylomicrons reach the circulation.14 At the same time, of Apo E and Apo C are transferred in the reverse direction from the HDL to chylomicrons.15 The Apo C interacts with lipoprotein lipase (LPL) to regulate triglyceride hydrolysis.16 Moreover, Apo E promotes the endocytic clearance of plasma lipoprotein by interacting with the LDL receptor-related protein, heparin sulfate proteoglycan, and other receptors involved in lipoprotein clearance.17 During circulation throughout the body, triglycerides are removed by the peripheral tissues. The endothelial-bound lipoprotein lipases that are located in capillaries of muscle and adipose tissue hydrolyze triglyceride to release fatty acid for muscle tissue to produce energy and for adipose tissue to store energy. During this process, Apo C detaches from the chylomicron. The hydrolyzed chylomicron or chylomicron remnant, containing Apo E and Apo B 48, are from circulation by the liver. The Apo E is required for the chylomicron clearance, while Apo C inhibits Apo E’s function to prevent removal of chylomicron.18 The Figure 1 below shows the apolipoprotein transfer between the lipoproteins.

Figure 1. Apolipoprotein Transfer between lipoproteins

The role of chylomicrons is to deliver dietary lipids to the peripheral organs, and the chylomicron would disappear within several hours after a meal. Therefore, the VLDL micelles are required to transport lipid in the absence of chylomicrons. VLDL is a triglyceride-enriched particle produced in the liver. Once nascent VLDLs, or Apo B 100 particles, are synthesized in the lumen of the endoplasmic reticulum (ER), they are transported to the Golgi, where these nascent VLDL micelles undergo a number of essential modifications. The mature VLDL particles then are transported to the plasma membrane and secreted into the circulatory system.19

The secretion of VLDL relies on the availability of triglyceride in the liver. Elevated hepatic triglyceride levels or fatty liver, increases VLDL and VLDL-Triglyceride secretion.20, 21 Once in circulation, newly synthesized VLDL takes up Apo C and Apo E from HDL after a few minutes. 22 The triglyceride content of VLDL is broken down into free fatty acids and glycerol rapidly by LPL and its cofactor, Apo C. After lipolysis, Apo C detaches from the smaller remnant of VLDL, termed IDL. The IDL is further hydrolyzed by hepatic lipase (HL) in

6 circulation. Since VLDL and its remnant (IDL) contain Apo C, some of the VLDL particles will never undergo complete dilipidation and may be removed from plasma as remnant particles.23

Ultimately, the dilipidated IDL (or LDL) is internalized by the interaction of Apo B 100 with

LDL receptor.24 The LDL receptor is present in the liver, muscle, adipose and other peripheral tissue, but the liver is the principal organ in clearing LDL via the LDL receptor (~70%) .25 If

LDL particles are oxidized, they can be removed by macrophage through the scavenger receptors.26 The lipoprotein metabolism is shown in the Figure 2.

Figure 2. Lipoprotein Metabolism

1.3 HDL & Reverse Cholesterol Transfer

High density lipoprotein (HDL) generally is known as ‘good’ cholesterol. In circulation,

HDLs are heterogeneous subpopulations of particles (micelles) that differ quantitatively and qualitatively in size, density, shape, apolipoprotein composition and lipid composition. 27

Compared to other lipoprotein micelles, HDLs have the highest relative density while being smallest in size.11 HDLs are involved in reverse cholesterol transport and act as a carrier of cholesterol back to the liver. Moreover, the concentration of cholesterol in the plasma HDL is generally inversely associated with the risk for the atherosclerosis development. The mechanism of HDL’s anti-atherosclerotic effect is complex and multifactorial. HDL has been demonstrated to exhibit beneficial effects on platelet function, endothelial function, coagulation parameters, inflammation, and interactions with triglyceride-rich lipoproteins. 28

HDLs originate as poorly lipidated apolipoproteins, termed Apo AI, secreted in the liver and small intestine. The liver secretes 70 to 80 percent of the total HDL in plasma, and the secretion rate is determined by genotype.29 A high dietary lipid absorption can increase Apo AI production in intestine30,31 The lipid-poor Apo AI undergo extensive modifications in the interstitial space and circulation and become mature HDL which contains additional apolipoproteins and lipids.32 After transport of lipid collected from circulation to the liver, mature HDLs would be converted back to Pre-HDL through hydrolysis by endothelial lipase

(EL) and hepatic lipase (HL). Pre-HDL can recycle multiple times in this pathway before being removal. Only poorly lipidated HDL is cleared by the kidney in the human body33

A significant function of HDL is the collection and delivery of cholesterol, which is not catabolized by non-hepatic cells, from peripheral sites such as the arterial wall, macrophages and smooth muscle cells back to the liver for reprocessing on excretion. The cholesterol delivery

9 is referred to as reverse cholesterol transport. 34 In circulation, newly synthesized Apo AI bind directly with high affinity to ATP-binding membrane cassette transport protein A1 (ABCA1), which is released from the liver or intestine, and found in the plasma membrane of cells.

ABCA1 translocases both free cholesterol and phospholipids from the outer shell on the Apo AI and convert Apo AI into discoidal particles, termed Pre-HDL. 35 Then Pre-HDL binds and reacts with the Lecithin cholesterol acyltransferase (LACT), which esterify free cholesterol in

HDL. The cholesterol ester produced by LCAT moves to the core of the discoidal HDL, converting the discoidal particles to spheres. The esterification of cholesterol by LCAT on HDL surface also maintains a concentration gradient of cholesterol, which allows HDL to absorb additional cholesterol by diffusion.

1.4 Lipase

There are three lipases, lipoprotein lipase (LPL), hepatic lipase (HL) and endothelial lipase (EL) involved in the lipid and lipoprotein metabolism. LPL catalyzes the hydrolysis of the triglyceride in circulating chylomicron and VLDL. The physiological site of LDL-medicated hydrolysis of lipoprotein is at the surface of the capillary endothelial cell in muscle and adipose tissue. LPL’s hydrolysis activity requires Apo C2 as its specific cofactor. 36 The hydrolysis reaction rate is proportional to the number of Apo C2 bound to Chylomicron or VLDL.37

Generally, more Apo C2 attach to chylomicrons than VLDLs.38

Hepatic lipase (HL) is a lipolytic enzyme primarily synthesized and secreted by the liver.

HL has triglyceridase and phospholipase activities. Thus, HL is involved in lipoprotein metabolism, particularly in HDL metabolism and triglyceride clearance of IDL and LDL.39 This generates smaller, denser HDLs and LDLs, and release l Apo AI for reverse cholesterol transport. 40

EL is in the same family as HL and LPL. EL is synthesized and secreted by endothelial cells and a variety of tissues. EL primarily hydrolyzes phospholipids in HDL and shows relatively little triglyceride lipase activity.41

1.5 CETP & PLTP

The lipid transfer protein (LTP), cholesterol ester transfer protein (CETP) and phospholipid transfer protein (PLTP), transfer triglyceride and phospholipid between HDL and other lipoproteins. CETP and PLTP are members of the lipopolysaccharide binding/lipid transfer protein family. Although PLTP and CETP have a moderate homology of sequence and similar structural features, they show no overlap in their in vivo functions.42

CETP mediates the transfer of cholesterol ester from HDL to LDL and VLDL, and of triglyceride from VLDL to HDL and LDL.43 The transfer of cholesterol ester and triglyceride are coupled reciprocally. The absolute lipid transfer rate is determined by plasma CETP concentration instead of donor and acceptor lipoprotein concentrations. The ratio of cholesterol ester to triglyceride transferred is dependent on donor lipoprotein composition. 44

Triglyceride transferred by CETP from VLDL results in triglyceride-enriched LDLs and

HDLs. Both triglyceride-enriched LDLs and HDLs are the preferred substrate for HL. HDLs would be hydrolyzed to reduce the particle size of HDL and release Apo AI. However, the hydrolysis of triglyceride-enriched LDLs produces small, dense LDLs, which are atherogenic.45

PLTP efficiently catalyzes the transfer of different lipids, including phospholipids, diacylglycerol, cerebrosides, and lipopolysaccharides between lipoproteins. Plasma PLTP mediates the net transfer of phospholipids from ApoB-containing-triglyceride-rich lipoprotein into HDL and also exchanges phospholipids between lipoproteins. In HDL metabolism, PLTP can transfer surface remnant, that is

11 composed of phospholipids and cholesterol, upon lipolysis of VLDL, IDL and chylomicron. Also, PLTP can convert small HDL particles into larger particles with an Apo AI. The PLTP-mediated conversion rate is proportional to the triglyceride content in small HDLs. 46,47

1.6 Small dense LDL

LDL has a significant role in the development and progression of atherosclerosis and cardiovascular disease. LDLs are diverse and generally divided into large LDL, intermediate

LDL and small dense LDL (sdLDL). The formation of sdLDL is seem primarily in the presence of a hypertriglyceridemic state. The hypertriglyceridemia is associated with elevated VLDL concentration and increased triglyceride transfer activity from triglyceride-rich lipoprotein to

LDLs and HDLs by CETP. The triglyceride-rich LDLs and HDLs are preferred substrates for hepatic lipase. After hydrolysis, triglyceride-rich LDLs are converted to sdLDLs.48

The sdLDL has a greater atherogenic potential than other LDL subclasses. LDL particles and remnants of chylomicron and VLDL are able to cross the endothelial barrier in blood vessels to reach the intima. During sub-endothelial retention, the interaction of Apo B 100-containing lipoprotein with proteoglycan in the extracellular matrix is thought to be the initial step in the development of atherogenesis. The sdLDL can penetrate the endothelial barrier 1.7 fold better than large LDL.49

The intimal proteoglycan is positively charged. Thus, electronegative LDL has atherogenic potential. Comparing to other LDLs, sdLDL has lower affinity to LDL receptor, which results in high retention time in circulation. Longer retention time increases the possibility of LDL modifications, including nonenzymatic glycosylation, NEFA enrichment, and modification by phospholipolytic enzymes. Those modifications result in high electronegativity

12 of LDL, lower affinity to LDL receptor and higher affinity to proteoglycan50,51 Thus sdLDL cholesterol is a much better marker for prediction of cardiovascular disease than total LDL cholesterol.

Modeling and Results

The modeling and results section is broken into two parts. In Part 1, an ordinary differential equation (ODE) model is generated to simulate the lipoprotein transport and lipid metabolism. In Part 2, the Gillespie algorithm is applied to simulate the transition of lipoprotein groups.

Chapter 2. Ordinary Differential Equation Model

In system biology, many biological processes can be modelled by reaction-rate equations expressing the rate of production of one species as a function of the concentration of other species in the system. An ODE model was developed to describe the system and approximate the evolution of lipoproteins by differential equations.

2.1 The lipoprotein species and transition

Lipoproteins can be divided into three groups depending on the apolipoproteins. The first class contains chylomicron (CM) and chylomicron remnant (CMR) that are synthesized and secreted from the intestine to delivery dietary lipid to the rest of the body. The second class includes VLDL, IDL, LDL-I (normal LDL), LDL-II (triglyceride-rich LDL) and LDL-

III(sdLDL). The third consists of Pre-HDL and HDL. Dietary fat in the intestine is packaged into

CM. Then CM goes into circulation and converted to CMR after hydrolysis in muscle and adipose tissue. Ultimately, the CMR is absorbed by the liver. The VLDL is secreted from the liver and is hydrolyzed by LPL in circulation. IDL is the remnant of VLDL, and LDL is produced when HL hydrolyzes IDL. Some LDL-I will obtain triglyceride through CETP from

VLDL, which results in the formation of LDL-II. The LDL-II can be hydrolyzed by HL to produce LDL_III. Pre-HDL is involved in return cholesterol from peripheral organs to the liver.

The concentrations of these lipoproteins are expressed as differential equation to track lipoprotein transition. The differential equations are summarized in the Appendix II.

2.2 Organ lipids and formation of lipoprotein

The concentrations of fatty acid, triglyceride and cholesterol in each organ is maintained by the lipoprotein transport system. Triglyceride and cholesterol are transported in lipoprotein micelles to peripheral organs and undergo hydrolysis by lipase or internalized via LDL receptor or LDL receptor-related proteins. Excess cholesterol and triglyceride released are taken up by

HDL, while low concentration free fatty acid can be delivered by blood directly. The muscle tissue is able to generate energy by utilizing glucose or fatty acid, but muscle prefers to utilize fatty acid when both glucose and fatty acid are sufficient. Adipose tissue decomposes triglyceride and releases fatty acid into the plasma through first order kinetics.

The intestine secretes CM and HDL, whose secretion rate is believed to be proportional to dietary lipid. The VLDL generation rate depends on hepatic triglyceride level and hepatic cholesterol level. The composition of newly synthesized VLDL depends on the ratio of hepatic triglyceride to hepatic cholesterol. 52 The conversion of VLDL to IDL requires the action of LPL, while the conversion of IDL to LDL requires the action of HL. The transitions of lipoprotein in circulation are based on first order kinetics.

2.3 Cholesterol ester transfer protein and sdLDL

CETP reciprocally transfers both cholesterol ester and triglycerides between lipoproteins, which includes VLDL, IDL and HDL in this model. With the action of CETP and HL, the LDLs are divided into LDL-I (normal LDL), LDL-II (triglyceride-rich LDL) and LDL-III (sdLDL).

HDLs are divided into Pre-HDL, HDL-I (normal HDL) and HDL-II (triglyceride-rich HDL).

Each type has different clearance rate and hydrolysis rate by HL. The CETP activity and lipoproteins are shown in Figure below.

Figure 3. CETP activity and LDLs & HDLs

2.4 LDL receptor & LDL receptor related protein

LDL receptor and LDL receptor-related protein are present in muscle, liver, adipose and other peripheral tissues. LDL can be fully absorbed into the tissue to release fatty acid and free cholesterol, whereas CMR, VLDL and IDL are only absorbed by adipose, muscle and liver tissue.

2.5 Results

The result of ODE model is composed of two parts. The first part is homeostasis of lipoprotein dynamic systems. The second part is the investigation of dietary intake’s effect on lipoprotein and lipid metabolism.

For homeostasis, the dietary lipid input is given through infusion or an intravenous bolus of chylomicron & HDL, whose amount depends on basal metabolism rate. Initially, infusion method was applied to achieve homeostasis state. Then the intravenous bolus of the same amount of chylomicron & HDL was given once per day to test the homeostasis of this dynamic system. The Figure 4 -8 below show the concentration change of lipoproteins or lipids.

Figure 4. CM and CMR concentration at homeostasis state

Figure 5. VLDL and IDL concentration at homeostasis state

Figure 6. LDL concentration at homeostasis state

Figure 7. HDL concentration at homeostasis state

Figure 8. Lipid concentration and cholesterol content in VLDL at homeostasis state

Although there are fluctuations of lipoprotein and lipid concentration through intravenous bolus method, the fluctuations are negligible. The VLDL cholesterol is tracked to observe the composition change of VLDL, IDL and LDL.

After increasing the daily dietary lipid intake by 25% percent, the excess lipid results in elevated concentrations of lipoproteins and lipids that are shown in Figure 9 - 14. The concentration CM, VLDL, IDL is increased around 20% after 25% increment of dietary lipid.

Whereas, the concentration increment percentage of LDL and HDL are smaller due to their large pool size.

Figure 9. CM and CMR concentration after increased dietary lipid intake

Figure 10. VLDL & IDL concentration after increased dietary lipid intake Because of the small pool size, the concentration of CM, VLDL and IDL achieve homeostasis in a short time. However, the LDL and HDL concentrations are still changing after

2880 hours following an intravenous bolus. In Figure 11, the concentrations of LDL-I and LDL-

II continue to rise long after the VLDL and LDL concentrations become constant. The LDL-III continues to rise over the entire within 120 days. The elevated concentrations of VLDL and

LDL-I promote CETP activity and a higher production rate of LDL-II. Then more LDL-II particles undergo hydrolysis of HL and are converted to LDL-III. Thus, excessive lipid intake would induce the liver to secret more VLDL, and more VLDL particles are transited to LDL, which results in an elevated level of sdLDL and higher risk for atherogenesis.

Figure 11. LDL concentration after increased dietary lipid intake

Figure 12. HDL concentration after increased dietary lipid intake

The higher dietary lipid absorption by the intestine leads to elevated Apo AI production rate. Also, the HDL cholesterol level, which is a biomarker for the primary prevention of atherosclerotic lesion’s development, is raised. Since Pre-HDL and HDL particles are anti- atherogenic, this suggests that dietary lipid intake should be sufficient to sustain a high HDL level.

Since basal metabolic rate in muscle tissue is lower than calories of lipid intake, triglyceride accumulates in adipose and muscle tissue. The levels of fatty acid in each organ are elevated as well, but the elevation is limited due to the low solubility of fatty acid in aqueous solution. The hepatic triglyceride is elevated, while hepatic cholesterol level is decreased.

(Figure 13) The lower ratio of hepatic cholesterol to hepatic triglyceride results in decreased cholesterol content in newly synthesized VLDL and average cholesterol content of circulating

VLDL particles. Thus, less free cholesterol is released from Apo B 100-containing particles.

Even though the cholesterol levels of muscle and adipose tissue are elevated initially, which is caused by high hydrolysis activity of VLDL, IDL and LDL, the cholesterol level drops after the concentration of VLDL and IDL achieve homeostasis.

Figure 13. Hepatic triglyceride & cholesterol after increased dietary lipid intake

Figure 14. lipid concentration and cholesterol content in VLDL after increased dietary lipid intake

If the pool size of ApoB 100 is small and limited, VLDL is generated based on Michaelis

Menten Kinetics. The Figure 15 -18 shows the lipoprotein and lipid concentration when Apo B

100 pool size is small. Since the Apo B 100 generation rate is limited, the level of VLDL, IDL and LDL are lower than in Figure 10 and Figure 11. The concentrations of HDL and HDL cholesterol are declining. The cholesterol concentration in each organ and average cholesterol content in VLDL particle is diminished as well. However, the hepatic triglyceride is rising.

Figure 15. VLDL & IDL concentration when VLDL generation is limited

Figure 16.LDL concentration when VLDL generation is limited

The limited VLDL secretion rate causes the accumulation of hepatic triglyceride, but cholesterol would be metabolized and cleared from the liver. Thus, the ratio of hepatic triglyceride level to hepatic concentration level ascends, which results in the synthesis of cholesterol-poor VLDL and less free cholesterol flux received by peripheral organs and HDL particles. Although LDL concentration is lower for people whose Apo B 100 pool is small, the low concentration of cholesterol may influence body regulation, since the cholesterol is an essential substrate for hormones and steroids. Moreover, the elevated hepatic triglyceride may result in fatty liver and related liver disease.

Figure 17. HDL concentration when VLDL generation is limited

Figure 18. Lipids concentration and average cholesterol content in VLDL when VLDL generation is limited

Chapter 3. Modified Gillespie algorithm:

Most models of fatty acid transport throughout the body rely on a system of differential equations to describe the lifetime of various lipoprotein micelles. These equations often sort the micelles into large, discrete groups, such as VLDL, IDL and LDL, with single-step transitions between the groups. However, in the body, this transition does not occur discretely, but rather through multiple interactions with enzymes. For example, in the transition from VLDL to IDL, a single lipoprotein can go through as many as ten microstates. In order to capture these nuances accurately, a system of differential equations would be too large to be practical. Additionally, stochasticity plays a large role in this process, and a system of differential equations would miss this. In order to solve these problems, we modelled fatty acid transport in the human body through a modified Gillespie algorithm.

3.1 Grouped Gillespie algorithm

The classic Gillespie algorithm simulates the path of one particle (micelle) at a time, at each state calculating all the other possible states and the rates of transition. Then, using the sum of the rates to construct an exponential distribution of possible rest times, it assigns a rest time in the current state and chooses the next state. After the rest time is complete, the molecule enters the next state, and the process repeats for the time specified. The algorithm then repeats itself for the specified number of particles. With fatty acid transport, since the states of the different lipoproteins also depend on the state of the entire system, we modified this strategy to create the

“grouped Gillespie algorithm.” Rather than tracking one particle at a time, this algorithm simulates the entire system at discrete time intervals, once again calculating the possible states for each lipoprotein at each step. However, rather than determining the rest time, the algorithm uses the exponential distribution to determine the probability of staying at the same state. Since

28 the exponential distribution is memoryless, the rest probability can be calculated the same way at each time step. Thus, as the system changes, each lipoprotein can update its state based on the system as a whole.

As a note, simulating the each of the of lipoprotein micelles in the entire human body would be computationally impractical. Therefore, each lipoprotein particle simulated actually represents many lipoprotein micelles, with the amount determined by the resolution of the model.

3.2 The three tracks

Lipoproteins in the body can be separated into three tracks based on the proteins that are found on their surface. The first consists of chylomicrons (CM) and chylomicron remnants

(CMR), which originate in the intestine and are eventually absorbed in the liver. The second consists of very low-density lipoprotein (VLDL), intermediate-density lipoprotein (IDL), and very low-density lipoprotein (LDL), which originate in the liver and are absorbed throughout the body. The third consists of high-density lipoprotein (HDL), which collects fatty acid from around the body and returns it to the liver. Fat from the diet first goes into the intestines, then

CM to CMR then is absorbed in the liver, where it forms VLDL and is released into circulation.

HDL is responsible for returning lipids, that not absorbed by organs, to the liver. The lipoproteins in each track follow a similar lifetime and thus can be grouped together.

3.3 Organ lipids and formation of lipoprotein

The model tracks the amount of fatty acid in each organ. Upon addition of fatty acid through one of the mechanisms described below, this value increases. The muscle tissue has a basal metabolic rate that decreases its value through zeroth order kinetics. The adipose tissue

29 releases its contents into the plasma, also through a zeroth order kinetics. The intestines release

CM, while the liver releases VLDL. These mechanisms are similar. A mean number of lipoproteins to be released is calculated based on the organ concentrations. The actual amount released is then determined based on a normal distribution, and the composition of each new lipoprotein formed also are based on normal distributions. The fatty acid released into lipoprotein are then deducted from the organ values. HDL is formed similarly from the action of HL.

The plasma delivers fatty acid back to the liver based on first order kinetics. Ideally, HDL would be formed from the plasma lipids for delivery back into the liver, but based on the amounts of fatty acid simulated, this would result in a very large amount of HDL needing to be simulated and thus a several-fold increase in computational time. However, since the amount of

HDL is comparatively so large, the first-order kinetics are a viable estimate.

3.4 Circulation:

Lipoprotein circulation through the body is modelled separately from the Gillespie algorithm. In the main (plasma) compartment, compartment time is determined through blood velocity and lipoprotein diffusion. After the time allotted, an organ is randomly chosen and organ time is calculated similarly. Afterwards, the lipoprotein is returned to the main compartment and the process repeats.

3.5 Lipoprotein lipase

CM transitions to CMR and VLDL transitions to IDL through lipoprotein lipase (LPL).

LPL is found in the muscle and adipose tissues and binds to ApoCII present on the surface of the lipoprotein. In our model, with each action of LPL, 21% of the triglycerides and 13% of the cholesteryl esters are hydrolyzed, which was determined from the number of Apo C lipoprotein

30 bound to CM & VLDL. The fatty acid is added to the tissue, and the ApoCII molecule is removed from the lipoprotein. CM is initialized with around 20 ApoCII molecules on their surface, while VLDL is initialized with 10. The transition from CM to CMR and VLDL to IDL is considered complete when there are no longer any ApoCII molecules attached to the lipoprotein. The only rate involved in this process is the attachment rate of LPL (k_LPL). The rest of the process is simulated instantaneously.

3.6 Hepatic lipase

Hepatic lipase (HL) governs the transition of IDL to LDL. In the model, hepatic lipase attaches to IDL with rate k_HL. Then, hydrolysis occurs under Michaelis-Menten kinetics.

Importantly, it is possible for more than one HL to bind an IDL. The off-rate (k_HLoff) determines how long hydrolysis is allowed to proceed. Once an HL detaches, as much of the hydrolyzed lipid as is allowed by HDL size constraints is added to an HDL, and the rest is released into the plasma.

3.7 Cholesteryl ester transfer protein

Cholesteryl ester transfer protein (CETP) transfers cholesteryl esters and triglycerides between lipoproteins, which can be VLDL, LDL, and HDL. The first rate associated with CETP is an attachment to the first lipoprotein (k_CETP1). Once a lipoprotein attaches to CETP, the next step is the other attachment of another lipoprotein with which to exchange lipids. The second rate is determined by the number of possible attachment candidates in the same organ

(k_CETP2). The smaller lipoprotein of the two gains TAG and loses CE.

3.8 LDL receptor

LDL receptor is present in muscle, liver, adipose, and peripheral tissue. Any of the lipoproteins in the VLDL track can be fully absorbed into tissue with LDL receptor, adding all of its lipids to that organ. CM and CMR can be similarly absorbed, but only in the liver. These rates are given by all the kmet variables in the model.

3.9 Result

Figure 19. Composition of each type of lipoprotein

The chylomicrons were simulated at about one tenth of their normal size to create more stochasticity. Their size goes down as LPL acts on them. The rest of the lipoproteins are regenerated, so their size remains fairly constant.

Figure 20. Body concentration through time

In accordance with Figure 20, there are immediate peak in CM, and a later peak in VLDL as the

CM are converted. VLDL then becomes IDL, and then LDL. Resulting in later and later peaks. Baseline values of the LDL species come from release from adipose tissue. (Numbers of lipoprotein here are the number simulated by the model. To obtain body concentration is just simple scalar multiplication.)

Figure 21. Organ composition.

Greater lipid is accumulated in adipose tissue than other tissues. The peak in muscle followed by decrease because of basal metabolic rate. Spikes in liver result from CM absorption. Likely, a simulation with more CM would lessen these spikes and result in more physiological accuracy. Plasma concentration stays fairly constant within the time scale.

3.10 Limitation

The greatest limitation from the model comes about from the size difference between CM and

CMR and VLDL. CM each have on the order of 106-7 lipid molecules, while VLDL contains only 103-4.

As a result, each CM creates around 1000 VLDL micelles. With the resources available, the computation time needed for simulating a diet that produces a minimal statistically preferable amount, about 100 CM, would be too large to be feasible on a regular computer. Additionally, we have simulated the transport of one meal. In the study of atherosclerosis, this type of information would need to span hundreds of meals, which would also be computationally infeasible. A supercomputer could be used to be able to simulate the system more accurately, but it might still take too much time. For that reason, this model could be used to help elucidate information on the transport mechanism, but for simulation of long-term phenomena, a different model would need to be used.

Another limitation is the lack of data. While we have obtained results that seem logical and are what we would expect, we have no data to evaluate them. To be able to judge our model results

34 accurately, pharmacokinetic profiles of the lipoproteins would be very useful. With this, our understanding of the overall mechanism can be greatly improved. The results detailed here come from model parameters that were mostly estimated based on model output, not from any physiological values.

In the future, it would be useful to attempt to link actual physiological values to the model for increased accuracy.

Chapter 4. Conclusion

The ODE model demonstrates the lipoprotein and lipid metabolism in a numerical way. The excessive lipid intake would result in the increased generation rates of VDL and LDL. The elevated

VLDL and LDL level promote CETP and higher formation of sdLDL production and heightened risk of atherosclerosis. Although this model does not include sugar metabolism for energy management, it still reveals the fatty acid transport within the human body.

The Gillespie algorithm model provides a new way to look at fatty acid transport. It has good expansion potential through new mechanisms such as oxidation of fatty acids and fatty acid types, as well as revision of old mechanisms. Additionally, if data is ever found against which the model can be compared, greater accuracy of the different parameters can be achieved. As it stands now, the model shows interesting trends and provides a template against which we can evaluate our understanding of fatty acid transport.

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Appendix

Appendix I Parameter for ODE

Volume Blood Flow rate Organ Volume (L) (L/min) Blood 5.9 5.348 liver 1.8 0.364 Muscle 30 0.952 fat 12.5 0.28 Other peripheral 12.3 3.752 Table 2. Volume and Blood Flow Rate of Organs53

Mole Chylomicron VLDL IDL LDL HDL CMR Triacylglycerol 6.18E-17 5.20E-20 4.92E-21 3.74E-22 1.59E-23 1.92E-19 Free Cholesterol 3.21E-18 1.54E-20 2.59E-21 1.12E-21 3.78E-23 1.39E-20 Cholesterol ester 2.89E-18 1.56E-20 5.01E-21 3.52E-21 8.82E-23 2.58E-20 Phospholipids 5.96E-18 1.99E-20 4.10E-21 1.54E-21 1.30E-22 8.66E-20 Concentration 7.81E-01 69 32 1272 35000 1.17E+01 nmol/L Total TG 2.91E-02 2.16E-03 9.48E-05 2.86E-04 3.35E-04 1.35E-03 Total FC 1.51E-03 6.40E-04 4.99E-05 8.58E-04 7.96E-04 9.75E-05 Total CE 1.36E-03 7.32E-06 2.36E-06 1.66E-06 4.15E-08 1.21E-05 Total Cholesterol 2.87E-03 6.48E-04 5.23E-05 8.59E-04 7.96E-04 1.10E-04 Table 3. Composition of lipoproteins in mole54–56

Water % Fat % (mass) Density Molarity Liver 68 15 1.05 1.87E-04 Adipose 10 90 0.95 9.18E-04 Muscle 76 5 1.05 5.64E-05 Other Peri 72 10 1.09 1.17E-04 Table 4. Triglyceride Concentration of Organs57-58

Cholesterol MUSCLE ADIPOSE skin Liver concentration g/L 19.58 19.24 10.02 1.76 molarity 1.69E-03 3.99E-03 2.11E-03 2.53E-03 Table 5. Cholesterol Concentration of Organs59-60

Molar C CLO(N=23) mass 12:00 0.19 639.02 121.41 20:00 0.12 975.67 117.08 20:4n-6 0.44 951.67 418.73 20:3n-3 0.05 957.67 47.88 20:5n-3 0.04 945.67 37.83 22:04 0.04 1,035.83 41.43 22:5n-3 0.18 1,029.83 185.37 1,023.83 194.53 22:6n-3 0.19 1,164.27 Sum 1.25 TG Molar mass 931.42 Table 6. Fatty Acid Profile and Triglyceride Molar Mass61

Appendix II Differential Equation & Matlab Code for ODE

Matlab Code for ODE

function dCondt = CholRbolus(t,Con,p) % parameters CMC = p(1); CMT = p(2); H1 = p(3); H2 = p(4); CMR = p(5); CMRR = p(6); H3 = p(7); H4 = p(8); VLDLR = p(9); VLDLG = p(10); IDLR = p(11); H5 = p (12); LDLR1 = p(13); LDLR2 = p(14); LDLR3 = p(15); LDLR4 = p(16); E1 = p(17); E2 = p(18); X1 = p(19); X2 = p(20); E3 = p(21); LDLR5= p(22); PREHDLI= p(23); HDLT= p(24); R4 = p(25); E6 = p(26); R1 = p(27); R2 = p(28); R3 = p(29); R5 = p(30); COG = p(31); ke = p(32); PREHDLE = p(33); CMRC = p(34); VLDLUC = p(35); IDLC = p(36); LDLC = p(37); LDLIC = p(38); LDLDC= p(39); PREHDLC = p(40); HDLC = p(41);

HDLIC = p(42); CMRT = p(43); IDLT = p(44); LDLT = p(45); LDLIT = p(46); LDLDT = p(47); PREHDLT = p(48); HDLT = p(49); HDLIT = p(50); VLDLT = p(51); T1 = p(52); T2 = p(53); T3 = p(54); T4 = p(55); TGG = p(56); KTGE = p(57); VLDLP = p(58); HDLP = p(59); LDLP = p(60); PREHDLLL = p(61); T5 = p(62); T6 = p(63); PI1= p(64); PI2= p(65);

KIN1= p(66); KIN2= p(67);

% Multicompartment

% VOlume V1 = 5.9; % (L) blood Volume V2 = 1.8; % (L) liver V3 = 30; % (L) Muscle V4 = 12.5; % (L) fat V5 = 73 - 10.5 - V1 -V2 -V3 - V4; % pther peripheral

% Bloof flow rate Q2 = 0.364*60; % Liver L/hr Q3 = 0.952*60; % Muscle L/hr Q4 = 0.280*60; % Fat L/hr Q5 = (5.348 - 0.280 - 0.364 - 0.952 -0.280)*60; % other peripehral

%%%%%% dCondt(1) =0; % cholesterol dCondt(2) =0; % fat

% lipoprotein CM concentration The secrection of CM is determined by FAtty Acid dCondt(3) =(Con(2)*10^-3-PREHDLI*Con(2)*HDLT)/CMT/V1-Q3/V1*(Con(3)-Con(4))... -Q4/V1*(Con(3)-Con(5))-Q2/V1*(Con(3)-Con(6)); % Plasma dCondt(4) =Q3/V3*(Con(3)-Con(4)) - H1*Con(4); % Muscle dCondt(5) =Q4/V4*(Con(3)-Con(5)) - H2*Con(5); % adipose dCondt(6) =Q2/V2*(Con(3)-Con(6))-CMR*Con(6); % Liver

% lipoprotein CMR concentration dCondt(7) = -Q3/V1*(Con(7)-Con(8)) - Q4/V1*(Con(7)-Con(9)) -Q2/V1*(Con(7)-Con(10)) ; % Plasma dCondt(8) =Q3/V3*(Con(7)-Con(8))+ H1*Con(4); % Muscle dCondt(9) =Q4/V4*(Con(7)-Con(9))+ H2*Con(5); % adipose dCondt(10)=Q2/V2*(Con(7)-Con(10)) - CMRR*Con(10); %liver

% lipoprotein VLDL The secrection of VLDL is determined by insulin and cholesterol in liver dCondt(11) =-Q3/V1*(Con(11)-Con(12))-Q4/V1*(Con(11)-Con(13))-Q2/V1*(Con(11)- Con(14)) ; % Plasma dCondt(12) =Q3/V3*(Con(11)-Con(12)) - H3*Con(12)-VLDLR*Con(12); % Muscle dCondt(13) =Q4/V4*(Con(11)-Con(13)) - H4*Con(13)-VLDLR*Con(13); % adipose dCondt(14) =Q2/V2*(Con(11)-Con(14)) + VLDLG*Con(37)*Con(56);% Con(43)*VLDLG*Con(27); % liver %dCondt(14) =Q2/V2*(Con(11)-Con(14)) + VLDLG; % liver

% lipoprotein IDL dCondt(15) =-Q3/V1*(Con(15)-Con(16))-Q4/V1*(Con(15)-Con(17))-Q2/V1*(Con(15)- Con(18)); % Plasma dCondt(16) =Q3/V3*(Con(15)-Con(16))+H3*Con(12)-IDLR*Con(16); % Muscle dCondt(17) =Q4/V4*(Con(15)-Con(17))+H4*Con(13)-IDLR*Con(17); % adipose dCondt(18) =Q2/V2*(Con(15)-Con(18)) -H5*Con(18) ; % liver

% lipoprotein LDL dCondt(19) =-Q3/V1*(Con(19)-Con(20))-Q4/V1*(Con(19)-Con(21))-Q2/V1*(Con(19)- Con(22))... -Q5/V1*(Con(19)-Con(23))-E1*Con(19)*Con(11); % Plasma dCondt(20) =Q3/V3*(Con(19)-Con(20))-LDLR2*Con(20); % Muscle dCondt(21) =Q4/V4*(Con(19)-Con(21))-LDLR3*Con(21); % adipose dCondt(22) =Q2/V2*(Con(19)-Con(22))+H5*Con(18)-Con(22)*LDLR1-E2*Con(22); % liver dCondt(23)= Q5/V5*(Con(19)-Con(23))-LDLR4*Con(23);% Other Peripheral

% LDL type II dCondt(24) =-Q3/V1*(Con(24)-Con(25))-Q4/V1*(Con(24)-Con(26))-Q2/V1*(Con(24)- Con(27))... -Q5/V1*(Con(24)-Con(28))+E1*Con(19)*Con(11); % Plasma dCondt(25) =Q3/V3*(Con(24)-Con(25))-LDLR2*X1*Con(25); % Muscle dCondt(26) =Q4/V4*(Con(24)-Con(26))-LDLR3*X1*Con(26); % adipose

55 dCondt(27) =Q2/V2*(Con(24)-Con(27))-LDLR1*X1*Con(27)-E3*Con(27); % liver dCondt(28)= Q5/V5*(Con(24)-Con(28))-LDLR4*X1*Con(28);% Other Peripheral

% LDL type III dCondt(29) =-Q3/V1*(Con(29)-Con(30))-Q4/V1*(Con(29)-Con(31))-Q2/V1*(Con(29)- Con(32))... -Q5/V1*(Con(29)-Con(33)); % Plasma dCondt(30) =Q3/V3*(Con(29)-Con(30))-LDLR2*X2*Con(30); % Muscle dCondt(31) =Q4/V4*(Con(29)-Con(31))-LDLR3*X2*Con(31); % adipose dCondt(32) =Q2/V2*(Con(29)-Con(32))+E2*Con(22)+E3*Con(27)-LDLR5*Con(32); % liver dCondt(33)= Q5/V5*(Con(29)-Con(33))-LDLR4*X2*Con(33);% Other Peripheral

%%%%% IDLC= VLDLC*0.245; LDLC= IDLC*0.611; IDLCR = 0.245; LDLCR = 0.611; X1 = 20.5/22.15;% diameter ratio of LDL-II to LDL-I X2 = 19/22.15; % diameter ratio of LDL-III to LDL-I % CHolesterol dCondt(34) =0; % Plasma dCondt(35) = (CMC-CMRC)*H1*Con(4)+(Con(64)- Con(64)*IDLCR)*H3*Con(12)+Con(64)*VLDLR*Con(12)...

+Con(64)*IDLCR*IDLR*Con(16)+Con(64)*IDLCR*LDLCR*LDLR2*Con(20)+Con(64)*IDL CR*LDLCR*(X1)^3*...

LDLR2*X1*Con(25)+Con(64)*IDLCR*LDLCR*(X2)^3*LDLR2*X2*Con(30)+(PREHDLC- HDLC)*Con(40)*R1*Con(35);% Muscle dCondt(36) = (CMC-CMRC)*H2*Con(5)+(Con(64)- Con(64)*IDLCR)*H4*Con(13)+Con(64)*VLDLR*Con(13)...

+Con(64)*IDLCR*IDLR*Con(17)+Con(64)*IDLCR*LDLCR*LDLR3*Con(21)+Con(64)*IDL CR*LDLCR*(X1)...

^3*LDLR3*X1*Con(26)+Con(64)*IDLCR*LDLCR*(X2)^3*LDLR3*X2*Con(31)+(PREHDL C-HDLC)*Con(41)*R3*Con(36);% adipose dCondt(37) = CMC*CMR*Con(6)+CMRC*CMRR*Con(10)-VLDLUC*Con(37)/2.53E- 03*1.69e-4/Con(56)*VLDL... G*Con(37)*Con(56)+(Con(64)*IDLCR- Con(64)*IDLCR*LDLCR)*H5*Con(18)+Con(64)*IDLCR*LDLCR*...

Con(22)*LDLR1+Con(64)*IDLCR*LDLCR*(X1)^3*LDLR1*X1*Con(27)+Con(64)*IDLCR* LDLCR*(X2)^3*...

LDLR5*Con(32)+(Con(64)*IDLCR*LDLCR- Con(64)*IDLCR*LDLCR*(X2)^3)*E2*Con(22)+(Con(64)*... IDLCR*LDLCR*(X1)^3-Con(64)*IDLCR*LDLCR*(X2)^3)*E3*Con(27)+(HDLC- PREHDLC)*Con(47)*R2+... (HDLIC-PREHDLC)*Con(52)*R4+COG-ke*Con(37);% liver dCondt(38) = Con(64)*IDLCR*LDLCR*LDLR4*Con(23)+Con(64)*IDLCR*LDLCR*(X1)^3*LDLR4*X1* Con(28)... +Con(64)*IDLCR*LDLCR*(X2)^3*LDLR4*X2*Con(33)+(PREHDLC- HDLC)*Con(43)*R5*Con(38);% Other Peripheral

% PREHDL dCondt(39) =-Con(39)*PREHDLE-Q3/V1*(Con(39)-Con(40))-Q4/V1*(Con(39)-Con(41))... -Q2/V1*(Con(39)-Con(42))-Q5/V1*(Con(39)-Con(43)); % Plasma dCondt(40) =Q3/V3*(Con(39)-Con(40))-Con(40)*R1*Con(35); % Muscle dCondt(41) =Q4/V4*(Con(39)-Con(41))-Con(41)*R3*Con(36); % adipose dCondt(42) =Q2/V2*(Con(39)-Con(42))+Con(47)*R2+Con(52)*R4+PREHDLLL; % liver dCondt(43)= Q5/V5*(Con(39)-Con(43))-Con(43)*R5*Con(38);% Other Peripheral

% HDL type I dCondt(44) =PREHDLI*Con(2)/V1-Q3/V1*(Con(44)-Con(45))-Q4/V1*(Con(44)-Con(46))... -Q2/V1*(Con(44)-Con(47))-Q5/V1*(Con(44)-Con(48))-E6*Con(44)*Con(11); % Plasma dCondt(45) =Q3/V3*(Con(44)-Con(45))+Con(40)*R1*Con(35); % Muscle dCondt(46) =Q4/V4*(Con(44)-Con(46))+Con(41)*R3*Con(36); % adipose dCondt(47) =Q2/V2*(Con(44)-Con(47))-Con(47)*R2; % liver dCondt(48)= Q5/V5*(Con(44)-Con(48))+Con(43)*R5*Con(38);% Other Peripheral

% HDL type II dCondt(49) =-Q3/V1*(Con(49)-Con(50))-Q4/V1*(Con(49)-Con(51))-Q2/V1*(Con(49)- Con(52))... -Q5/V1*(Con(49)-Con(53))+E6*Con(44)*Con(11); % Plasma dCondt(50) =Q3/V3*(Con(49)-Con(50)); % Muscle dCondt(51) =Q4/V4*(Con(49)-Con(51)); % adipose dCondt(52) =Q2/V2*(Con(49)-Con(52))-Con(52)*R4; % liver dCondt(53)= Q5/V5*(Con(49)-Con(53));% Other Peripheral

% Triglyceride

TGM = 5.64e-5; dCondt(54) =(PREHDLT-HDLT)*Con(40)*R1*Con(35) + T1*Con(59)/3 - T5*Con(54)/(Con(54)+5.64e-6*10) ... +VLDLT*VLDLR*Con(12)+IDLT*IDLR*Con(1)+LDLT*LDLR2*Con(20)+... LDLIT*LDLR2*X1*Con(25)+LDLDT*LDLR2*X2*Con(30) - (KTGE*1.15)*(Con(54)/(Con(54)+TGM*0.15));% Muscle dCondt(55) =(PREHDLT-HDLT)*Con(41)*R3*Con(36) + T2*Con(60)/3 - T6*Con(55)/(Con(55)+9.18e-5*0.1)... +VLDLT*VLDLR*Con(13)+IDLT*IDLR*Con(17)+LDLT*LDLR3*Con(21)+... LDLIT*LDLR3*X1*Con(26)+LDLDT*LDLR3*X2*Con(31) ; % adipose %+T2*Con(46)/3 dCondt(56) =-VLDLT*VLDLG*Con(37)*Con(56)+TGG -T3*Con(56) ... +CMT*CMR*Con(6)+CMRT*CMRR*Con(10)... +LDLT*Con(22)*LDLR1+LDLIT*LDLR1*X1*Con(27)+LDLDT*LDLR5*Con(32);% liver dCondt(57)= (PREHDLT-HDLT)*Con(43)*R5*Con(38) - T4*Con(57) ... +LDLT*LDLR4*Con(23)+LDLIT*LDLR4*X1*Con(28)... +LDLDT*LDLR4*X2*Con(33);% Other Peripheral

% Fatty Acid dCondt(58) =-Q3/V1*(Con(58)-Con(59))-Q4/V1*(Con(58)-Con(60))-Q2/V1*(Con(58)-... Con(61))-Q5/V1*(Con(58)-Con(62)); % Plasma

dCondt(59) = Q3/V3*(Con(58)-Con(59)) - T1*Con(59) + T5*Con(54)/(Con(54)+5.64e-6*10)*3 ... +3*( (CMT-CMRT)*H1*Con(4)+(VLDLT-IDLT)*H3*Con(12)) ; % Muscle dCondt(60) = Q4/V4*(Con(58)-Con(60)) - T2*Con(60) + T6*Con(55)/(Con(55)+9.18e- 5*0.1)*3 ... +3*((CMT-CMRT)*H2*Con(5)+(VLDLT-IDLT)*H4*Con(13));% adipose dCondt(61) = Q2/V2*(Con(58)-Con(61)) + T3*Con(56)*3 ... +3*((IDLT-LDLT)*H5*Con(18)+(LDLT-LDLDT)*E2*Con(22)+(LDLIT- LDLDT)*E3*Con(27)... +(HDLT-PREHDLT)*Con(47)*R2+(HDLIT-PREHDLT)*Con(52)*R4) ; % liver dCondt(62) = Q5/V5*(Con(58)-Con(62)) + T4*Con(57)*3 ; % Other Peripheral

dCondt(63)=(VLDLG*Con(37)*Con(56)*V2- ((H4+VLDLR)*Con(13)*V4+(H3+VLDLR)*Con(12)*V3))... /(Con(11)*(V1+V5)+Con(12)*V3+Con(13)*V4+Con(14)*V2);%count lipoprotein change dCondt(64) = (VLDLG*Con(37)*Con(56)*V2*VLDLUC*Con(37)/2.53E-03*1.69e-4/Con(56)- ... Con(64)*((H4+VLDLR)*Con(13)*V4+(H3+VLDLR)*Con(12)*V3)+(HDLC- HDLIC)*E6*Con(44)... *Con(11)+(Con(64)*IDLCR*LDLCR- Con(64)*IDLCR*LDLCR*(X1)^3)*E1*Con(19)*Con(11))... /(Con(11)*(V1+V5)+Con(12)*V3+Con(13)*V4+Con(14)*V2);% VLDL-C

dCondt(65) = (Con(19)+Con(24))*PI1 - KIN1*Con(65); dCondt(66) = Con(29)*PI2 - KIN2*Con(66); %intima dCondt = dCondt';

clear all; clc; format long % Multicompartment

% VOlume V1 = 5.9; % (L) blood Volume V2 = 1.8; % (L) liver V3 = 30; % (L) Muscle V4 = 12.5; % (L) fat V5 = 73 - 10.5 - V1 -V2 -V3 - V4; % pther peripheral

% Bloof flow rate Q2 = 0.364*60; % Liver L/hr Q3 = 0.952*60; % Muscle L/hr Q4 = 0.280*60; % Fat L/hr Q5 = (5.348 - 0.280 - 0.364 - 0.952 -0.280)*60; % other peripehral

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%% % input parameter KC = 0.259/24; %mmol KF = 128.1/24; %mmol

Cholesterol=KC; Fat =KF;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%

% TG in lipoprotein CMT = 6.18E-17; CMRT = CMT*3.10E-03; VLDLT = 5.20E-20*2; IDLT = VLDLT*0.095; LDLT = IDLT*0.076; HDLT = 1.59E-23; PREHDLT = 0; LDLIT = LDLT; LDLDT = LDLT*(19/22.15)^3; HDLIT= HDLT; % Cholesterol Molecule Concentration COP = 0.5*10^-3; %mol/L COM = 1.69E-03; COA = 3.99E-03; COL= 2.53E-03; COO = 2.11E-03;

PREHDLI = 2.1424e+17/3/KF;

% Choloesterol in lipoprotein HDLC= 1.26E-22; PREHDLC= 0; CMC = (Cholesterol*10^-3+KC/24*10^-3-PREHDLI*KF*HDLC)/(KF/CMT); % CMC= (Cholesterol/(Fat/CMT)+6.11E-18*(CMP*V1+CMM*V3+CMA*V4+CML*V2+CMP*V5))/... % (Fat/CMT+CMP*V1+CMM*V3+CMA*V4+CML*V2+CMP*V5); CMRC=CMC*6.49E-03; VLDLC= 3.10E-20; IDLC= VLDLC*0.245; LDLC= IDLC*0.611; LDLIC = LDLC*(20.5/22.15)^3; LDLDC = LDLC*(19/22.15)^3; HDLIC = HDLC* (8.8/11)^3;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%

CMP = 12.5e-9*6.02*10^23*.06;%nmol/L () % CMR = 5e-5; CML = (-(KF-PREHDLI*KF*HDLT*1000) + 1000*CMT*CMP*Q2 + 1000*CMT*CMP*Q3 +... 1000*CMT*CMP*Q4)/(1000*CMT*(Q2 + Q3 + Q4)); CMM = (-(KF-PREHDLI*KF*HDLT*1000) + 1000*CMT*CMP*Q2 + 1000*CMT*CMP*Q3 +... 1000*CMT*CMP*Q4)/(1000*CMT*(Q2 + Q3 + Q4)); CMA = (-(KF-PREHDLI*KF*HDLT*1000) + 1000*CMT*CMP*Q2 + 1000*CMT*CMP*Q3 +... 1000*CMT*CMP*Q4)/(1000*CMT*(Q2 + Q3 + Q4));

H1 =((CMP - CMM)*Q3)/(CMM*V3); H2 =((CMP - CMA)*Q4)/(CMA*V4); CMR = ((CMP - CML)*Q2)/(CML*V2);

CMRP = 12.5e-9*6.02*10^23*.94; CMRM = (CMRP*Q3 + CMM*H1*V3)/Q3; CMRA = (CMRP*Q4 + CMA*H2*V4)/Q4; CMRL = (CMRP*Q2 + CMRP*Q3 - CMRM*Q3 + CMRP*Q4 - CMRA*Q4)/Q2; CMRR = -(((CMRL - CMRP)*Q2)/(CMRL*V2));

VLDLP =69e-9*6.02*10^23; %VLDLP

% dietary cholesterol : self cholesterol synthesis

TGHP = 1.690000000000000e-04; VLDLG = (CMRR*CMRL*CMRT+CMR*CML*CMT)/(VLDLT-0.3*LDLT)/TGHP/COL; % VLDLR = 0.000001; VLDLL = (VLDLP*Q2 + V2*VLDLG*TGHP*COL)/Q2;%VLDLL VLDLM = (VLDLP*Q2 - VLDLL*Q2 + VLDLP*Q3 + VLDLP*Q4)/(Q3 + Q4); %VLDLM VLDLA = (VLDLP*Q2 - VLDLL*Q2 + VLDLP*Q3 + VLDLP*Q4)/(Q3 + Q4);% VLDLA

H3 =(VLDLP*Q3 - VLDLM*Q3 - VLDLM*V3*VLDLR)/(VLDLM*V3); H4 =(VLDLP*Q4 - VLDLA*Q4 - VLDLA*V4*VLDLR)/(VLDLA*V4);

IDLR = VLDLR; IDLP = 32e-9*6.02*10^23; IDLM = (IDLP*Q3 + VLDLM*H3*V3)/(Q3 + IDLR*V3); IDLA = (IDLP*Q4 + VLDLA*H4*V4)/(Q4 + IDLR*V4); IDLL = (IDLP*Q2 + IDLP*Q3 - IDLM*Q3 + IDLP*Q4 - IDLA*Q4)/Q2; H5 = ((IDLP - IDLL)*Q2)/(IDLL*V2);

LDLP = 1272e-9*6.02*10^23*0.6; % Solve[Q2/V2*(LDLP - LDLL) + H5*IDLL*0.3 == 0, LDLL] LDLL = ((1.*(IDLL*H5-0.6*IDLL*H5 + (LDLP*Q2)/V2)*V2)/Q2); E2 = IDLL*H5*0.04261/LDLL; LDLR1 = (LDLP*Q2 - LDLL*Q2 - LDLL*E2*V2 + IDLL*H5*V2)/(LDLL*V2); E1 = IDLL*H5*0.1/LDLP/VLDLP;%??????? LDLM = (LDLP*Q2 - LDLL*Q2 + LDLP*Q3 + LDLP*Q4 + LDLP*Q5 +VLDLP*LDLP*E1*V1)/(Q3 + Q4 + Q5); LDLA = (LDLP*Q2 - LDLL*Q2 + LDLP*Q3 + LDLP*Q4 + LDLP*Q5 +VLDLP*LDLP*E1*V1)/(Q3 + Q4 + Q5); LDLO = (LDLP*Q2 - LDLL*Q2 + LDLP*Q3 + LDLP*Q4 + LDLP*Q5 +VLDLP*LDLP*E1*V1)/(Q3 + Q4 + Q5);

LDLR2 = ((LDLP - LDLM)*Q3)/(LDLM*V3); LDLR3 = ((LDLP - LDLA)*Q4)/(LDLA*V4); LDLR4 = ((LDLP - LDLO)*Q5)/(LDLO*V5); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%% % http://www.lipo-search.com/eng/clients/clinical.php

X1 = 20.5/22.15;

LDLIP = 1272e-9*6.02*10^23*0.15; LDLIM = (LDLIP*Q3)/(Q3 + LDLR2*V3*X1); LDLIA = (LDLIP*Q4)/(Q4 + LDLR3*V4*X1); LDLIO = (LDLIP*Q5)/(Q5 + LDLR4*V5*X1); LDLIL = (LDLIP*Q2 + LDLIP*Q3 - LDLIM*Q3 + LDLIP*Q4 - LDLIA*Q4 + LDLIP*Q5 -... LDLIO*Q5 - VLDLP*LDLP*E1*V1)/Q2; E3 = (LDLIP*Q2 - LDLIL*Q2 - LDLIL*LDLR1*V2*X1)/(LDLIL*V2);

X2 = 19/22.15; LDLDP = 1272e-9*6.02*10^23*0.25; LDLDM = (LDLDP*Q3)/(Q3 + LDLR2*V3*X2); LDLDA = (LDLDP*Q4)/(Q4 + LDLR3*V4*X2); LDLDO= (LDLDP*Q5)/(Q5 + LDLR4*V5*X2); LDLDL = (LDLDP*Q2 + LDLDP*Q3 - LDLDM*Q3 + LDLDP*Q4 - LDLDA*Q4 + LDLDP*Q5 -... LDLDO*Q5)/Q2; LDLR5 = (LDLDP*Q2 - LDLDL*Q2 + LDLL*E2*V2 + LDLIL*E3*V2)/(LDLDL*V2);

(Q2/V2*(LDLDP-LDLDL)+E2*LDLL+E3*LDLIL)/LDLDL X4 = LDLR5/LDLR1 -X2

X=0; y=0; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%

ke = (KC*10^-3)*2/(V2)/24/(COL*10^-3); PREHDLP = 24000e-9*6.02*10^23;

HDLGM = (1/((HDLC - PREHDLC)))*((CMC-CMRC)*H1*CMM+(VLDLC-IDLC)*H3*VLDLM+...

VLDLC*VLDLR*VLDLM+IDLC*IDLR*IDLM+LDLC*LDLR2*LDLM+LDLIC*LDLR2*X1*LDLIM+LDLDC *LDLR2*X2*LDLDM);

HDLGA = (1/((HDLC - PREHDLC)))*((CMC-CMRC)*H2*CMA+(VLDLC-IDLC)*H4*VLDLA+...

VLDLC*VLDLR*VLDLA+IDLC*IDLR*IDLA+LDLC*LDLR3*LDLA+LDLIC*LDLR3*X1*LDLIA+LDLDC* LDLR3*X2*LDLDA);

HDLGO = (LDLC*LDLR4*LDLO+LDLIC*LDLR4*X1*LDLIO+LDLDC*LDLR4*X2*LDLDO)/((HDLC- PREHDLC));

PREHDLM = (PREHDLP*Q3 - HDLGM*V3)/Q3; PREHDLA = (PREHDLP*Q4 - HDLGA*V4)/Q4; PREHDLO = (PREHDLP*Q5 - HDLGO*V5)/Q5;

R1 = ((PREHDLP - PREHDLM)*Q3)/(PREHDLM*V3)/COM; R3 = ((PREHDLP - PREHDLA)*Q4)/(PREHDLA*V4)/COA; R5 = ((PREHDLP - PREHDLO)*Q5)/(PREHDLO*V5)/COO;

PREHDLE = 2.1424e+17/V1/PREHDLP; PREHDLLL = 2.1424e+17*2/3/V2; PREHDLL =(PREHDLP*Q2 + PREHDLP*Q3 - PREHDLM*Q3 + PREHDLP*Q4 - PREHDLA*Q4 + PREHDLP*Q5 -... PREHDLO*Q5 + PREHDLP*PREHDLE*V1)/Q2;

HDLP = 8000e-9*6.02*10^23; HDLM =(HDLP*Q3 + COM*PREHDLM*R1*V3)/Q3; HDLA =(HDLP*Q4 + COA*PREHDLA*R3*V4)/Q4; HDLO =(HDLP*Q5 + COO*PREHDLO*R5*V5)/Q5;

E6 =(PREHDLI*KF/V1*13)/(VLDLP*HDLP);

HDLL=-(PREHDLI*KF/V1-Q3/V1*(HDLP-HDLM)-Q4/V1*(HDLP-HDLA)... -Q2/V1*HDLP-Q5/V1*(HDLP-HDLO)-E6*HDLP*VLDLP)/(Q2/V1);

HDLIP = 3000e-9*6.02*10^23;

HDLIM = HDLIP; HDLIA = HDLIP; HDLIO = HDLIP;

HDLIL = (HDLIP*Q2 + HDLIP*Q3 - HDLIM*Q3 + HDLIP*Q4 - HDLIA*Q4 + HDLIP*Q5 -... HDLIO*Q5 - VLDLP*HDLP*E6*V1)/Q2;

R4 = ((HDLIP - HDLIL)*Q2)/(HDLIL*V2); R2 =(-PREHDLP*Q2 + PREHDLL*Q2 - HDLIL*R4*V2-PREHDLLL*V2)/(HDLL*V2);%((HDLP - HDLL)*Q2)/(HDLL*V2); X3 = R4/R2;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%

VLDLUCi= 3.013017788480899e-20;

COG =-(CMC*CMR*CML+CMRC*CMRR*CMRL-VLDLUCi*VLDLG*TGHP*COL+(IDLC- LDLC)*H5*IDLL... +LDLC*LDLL*LDLR1+LDLIC*LDLR1*X1*LDLIL+LDLDC*LDLR5*LDLDL+(LDLC- LDLDC)*E2*LDLL+(LDLIC-LDLDC)*E3*LDLIL+... (HDLC-PREHDLC)*HDLL*R2+(HDLIC-PREHDLC)*HDLIL*R4-ke*COL);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% FAP = 474.6e-6; TGM = 5.64e-5; TGA = 9.18e-4; TGL= 1.69e-4; TGO = 1.17e-4;

KTGE = KF/(V3)*10^-3;

% FAOO = 4.746024511192042e-04;

%FAO = ((3*COO*PREHDLO*(HDLT - PREHDLT)*R5)/T4); AFA =6.851127525125914e-04; MFA = 4.047815140347522e-04;

T4 = ((PREHDLT-HDLT)*PREHDLO*R5*COO ... +LDLT*LDLR4*LDLO+LDLIT*LDLR4*X1*LDLIO... +LDLDT*LDLR4*X2*LDLDO)/TGO;

T22 = -3*((PREHDLT-HDLT)*PREHDLA*R3*COA ... +VLDLT*VLDLR*VLDLA+IDLT*IDLR*IDLA+LDLT*LDLR3*LDLA+... LDLIT*LDLR3*X1*LDLIA+LDLDT*LDLR3*X2*LDLDA)/AFA;

T11 = -3*((PREHDLT-HDLT)*PREHDLM*R1*COM ... +VLDLT*VLDLR*VLDLM+IDLT*IDLR*IDLM+LDLT*LDLR2*LDLM+... LDLIT*LDLR2*X1*LDLIM+LDLDT*LDLR2*X2*LDLDM - KTGE)/MFA;

T5 = T11*MFA*0.0001/(TGM*3); T6 = T22*AFA*0.05/(TGA*3);

T2 = -3*((PREHDLT-HDLT)*PREHDLA*R3*COA ... +VLDLT*VLDLR*VLDLA+IDLT*IDLR*IDLA+LDLT*LDLR3*LDLA+... LDLIT*LDLR3*X1*LDLIA+LDLDT*LDLR3*X2*LDLDA - T6*TGA/(TGA+9.18e-5*0.1))/AFA;

T1 = -3*((PREHDLT-HDLT)*PREHDLM*R1*COM ... +VLDLT*VLDLR*VLDLM+IDLT*IDLR*IDLM+LDLT*LDLR2*LDLM+... LDLIT*LDLR2*X1*LDLIM+LDLDT*LDLR2*X2*LDLDM - KTGE - T5*TGM/(TGM+5.64e-6*10))/MFA;

FAO = (FAP*Q5 + 3*T4*TGO*V5)/Q5;

FAM = (1/Q3)*(FAP*Q3 - 3*CMRT*CMM*H1*V3 + 3*CMT*CMM*H1*V3 -... 3*VLDLM*H3*IDLT*V3 - T1*MFA*V3 +3*T5*TGM/(TGM+5.64e-6*10)*V3 + 3*VLDLM*H3*V3*VLDLT);

FAA =(FAP*Q4 - 3*CMRT*CMA*H2*V4 + 3*CMT*CMA*H2*V4 -... 3*VLDLA*H4*IDLT*V4 - T2*AFA*V4 + 3*T6*TGA/(TGA+9.18e-5*0.1)*V4+ 3*VLDLA*H4*V4*VLDLT)/Q4;

FAL = (FAP*Q2 + FAP*Q3 - FAM*Q3 + FAP*Q4 - FAA*Q4 + FAP*Q5 -... FAO*Q5)/Q2;

T3 = -1/3*(Q2/V2*(FAP-FAL) ... +3*((IDLT-LDLT)*H5*IDLL+(LDLT-LDLDT)*E2*LDLL+(LDLIT-LDLDT)*E3*LDLIL... +(HDLT-PREHDLT)*HDLL*R2+(HDLIT-PREHDLT)*HDLIL*R4))/TGL;

TGG =0;

% -(-VLDLT*VLDLG*COL*TGL-T3*TGL ... % +CMT*CMR*CML+CMRT*CMRR*CMRL... % +LDLT*LDLL*LDLR1+LDLIT*LDLR1*X1*LDLIL+LDLDT*LDLR5*LDLDL); %

GGG=(FAL*T3/(3*COL*TGL*VLDLT));

INTIMA1 = 60; KIN1 = 51/(INTIMA1); PI1 = KIN1*(INTIMA1)/ (LDLP+LDLIP);

PI2 = PI1*1.7; KIN2 = KIN1*3; INTIMA2 =LDLDP*PI2/KIN2 ;

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Z = 0; VLDLUC = VLDLC-((HDLC-HDLIC)*E6*HDLP*VLDLP+(LDLC- LDLIC)*E1*LDLP*VLDLP)/(VLDLG*TGL*COL*V2);

Co =[Cholesterol Fat... CMP CMM CMA CML... CMRP CMRM CMRA CMRL... VLDLP VLDLM VLDLA VLDLL... IDLP IDLM IDLA IDLL... LDLP LDLM LDLA LDLL LDLO... LDLIP LDLIM LDLIA LDLIL LDLIO... LDLDP LDLDM LDLDA LDLDL LDLDO... COP COM COA COL COO... PREHDLP PREHDLM PREHDLA PREHDLL PREHDLO... HDLP HDLM HDLA HDLL HDLO... HDLIP HDLIM HDLIA HDLIL HDLIO... TGM TGA TGL TGO... FAP FAM FAA FAL FAO... Z VLDLC ... INTIMA1 INTIMA2 ];... p = [CMC CMT H1 H2 CMR CMRR... H3 H4 VLDLR VLDLG... IDLR H5... LDLR1 LDLR2 LDLR3 LDLR4... E1 E2 X1 X2 E3 LDLR5... PREHDLI HDLT R4 E6... R1 R2 R3 R5 COG ke PREHDLE... CMRC VLDLUC IDLC LDLC LDLIC LDLDC... PREHDLC HDLC HDLIC... CMRT IDLT LDLT LDLIT LDLDT... PREHDLT HDLT HDLIT VLDLT... T1 T2 T3 T4 TGG KTGE... VLDLP HDLP LDLP PREHDLLL T5 T6... PI1 PI2 KIN1 KIN2 ]; options = odeset('MaxStep',5e-2, 'AbsTol', 1e-10,'RelTol', 1e-12,'InitialStep', 1e-2); %%%%%%%%%%%%%%%%%%%%%%%%%*************************************** q=1.2; % infusion timespan = 1; [t0,C0] = ode45(@CholRbolus,[0 timespan],Co,options,p); timespanI = timespan+24*121; Co1 = C0(length(C0(:,1)),:);

Co1(1,1) = 0.259/24*q; %cholesterol since CMT is constant Co1(1,2) = 128.1/24*q; %fat p(:,1) = CMC;

[ts,Cs] = ode45(@CholRbolus,[timespan timespanI],Co1,options,p); tI = [t0;ts]; CI = [C0;Cs];

% %%%%%%%%%%%%%%%%%%%%%%%%%************************% %figure PREHDL;

%bolus

timespan1 = timespan+12; Co1 = C0(length(C0(:,1)),:); Co1(1,1) = 0; Co1(1,2) = 0; % Co1(1,3) = C0(length(C0(:,3)),3)+(KF*10^-3-PREHDLI*KF*HDLT)/CMT/V1*24*q; % Co1(1,44) = C0(length(C0(:,44)),44)+24*PREHDLI*KF/V1*q; p(:,1) = CMC*1; [t1,C1] = ode45(@CholRbolus,[timespan timespan1],Co1,options,p); t = [t0;t1]; C = [C0;C1]; Co2 = C1(length(C1(:,1)),:); timespan2 = timespan1 +24; Co2(1,3) = C1(length(C1(:,3)),3)+(KF*10^-3-PREHDLI*KF*HDLT)/CMT/V1*24*q; Co2(1,44) = C1(length(C1(:,44)),44)+24*PREHDLI*KF/V1*q; for i = 1:120 [t2,C2] = ode45(@CholRbolus,[timespan1 timespan2],Co2,options,p); t = [t;t2]; C = [C;C2]; Co2 = C2(length(C2(:,1)),:); Co2(1,3) = C2(length(C2(:,3)),3)+(KF*10^-3-PREHDLI*KF*HDLT)/CMT/V1*24*q; Co2(1,44) = C2(length(C2(:,44)),44)+24*PREHDLI*KF/V1*q; timespan1 = timespan1 +24; timespan2 = timespan2 +24; i = i + 1; end

% %figure CM; subplot(2,2,1) hold on plot(t,C(:,3),... 'g','LineWidth',1) plot(t,C(:,4),... 'm','LineWidth',1) plot(t,C(:,5),... 'b','LineWidth',1)

67 plot(t,C(:,6),... 'r','LineWidth',1) % plot(tI,CI(:,3),... % 'g','LineWidth',1) % plot(tI,CI(:,4),... % 'm','LineWidth',1) % plot(tI,CI(:,5),... % 'b','LineWidth',1) % plot(tI,CI(:,6),... % 'r','LineWidth',1) title('Chylomicrons'); xlabel('Time hr'); ylabel('Concentration #'); legend('Plasma','Muscle', 'Fat','Liver');

%figure CMR; subplot(2,2,2) hold on plot(t,C(:,7),... 'g') plot(t,C(:,8),... 'm') plot(t,C(:,9),... 'b') plot(t,C(:,10),... 'r') % plot(tI,CI(:,7),... % 'g') % plot(tI,CI(:,8),... % 'm') % plot(tI,CI(:,9),... % 'b') % plot(tI,CI(:,10),... % 'r') title('Chylomicrons Remnant'); xlabel('Time hr'); ylabel('Concentration #'); legend('Plasma','Muscle', 'Fat','Liver'); subplot(2,2,3) hold on plot(t,C(:,3)+C(:,7),... 'g','LineWidth',1) plot(t,C(:,4)+C(:,8),... 'm','LineWidth',1) plot(t,C(:,5)+C(:,9),... 'b','LineWidth',1) plot(t,C(:,6)+C(:,10),... 'r','LineWidth',1) % plot(tI,CI(:,3)+CI(:,7),... % 'g','LineWidth',1) % plot(tI,CI(:,4)+CI(:,8),... % 'm','LineWidth',1)

% plot(tI,CI(:,5)+CI(:,9),... % 'b','LineWidth',1) % plot(tI,CI(:,6)+CI(:,10),... % 'r','LineWidth',1) title('Apo B48'); xlabel('Time hr'); ylabel('Concentration #'); legend('Plasma','Muscle', 'Fat','Liver');

% figure VLDL; subplot(2,1,1) hold on plot(t,C(:,11),... 'g') plot(t,C(:,12),... 'm') plot(t,C(:,13),... 'b') plot(t,C(:,14),... 'r') % plot(tI,CI(:,11),... % 'g') % plot(tI,CI(:,12),... % 'm') % plot(tI,CI(:,13),... % 'b') % plot(tI,CI(:,14),... % 'r') title('VLDL'); xlabel('Time hr'); ylabel('Concentration #'); legend('Plasma','Muscle', 'Fat','Liver');

% figure IDL; subplot(2,1,2) hold on plot(t,C(:,15),... 'g') plot(t,C(:,16),... 'm') plot(t,C(:,17),... 'b') plot(t,C(:,18),... 'r') % plot(tI,CI(:,15),... % 'g') % plot(tI,CI(:,16),... % 'm') % plot(tI,CI(:,17),... % 'b') % plot(tI,CI(:,18),... % 'r') title('IDL'); xlabel('Time hr');

69 ylabel('Concentration #'); legend('Plasma','Muscle', 'Fat','Liver');

% figure LDL-I; subplot(2,2,1) hold on plot(t,C(:,19),... 'g') plot(t,C(:,20),... 'm') plot(t,C(:,21),... 'b') plot(t,C(:,22),... 'r') plot(t,C(:,23),... 'k') % plot(tI,CI(:,19),... % 'g') % plot(tI,CI(:,20),... % 'm') % plot(tI,CI(:,21),... % 'b') % plot(tI,CI(:,22),... % 'r') % plot(tI,CI(:,23),... % 'k') title('LDL-I'); xlabel('Time hr'); ylabel('Concentration #'); legend('Plasma','Muscle', 'Fat','Liver','Other Peri');

% figure LDL-II; subplot(2,2,2) hold on plot(t,C(:,24),... 'g') plot(t,C(:,25),... 'm') plot(t,C(:,26),... 'b') plot(t,C(:,27),... 'r') plot(t,C(:,28),... 'k') % plot(tI,CI(:,24),... % 'g') % plot(tI,CI(:,25),... % 'm') % plot(tI,CI(:,26),... % 'b') % plot(tI,CI(:,27),...

% 'r') % plot(tI,CI(:,28),... % 'k') title('LDL-II'); xlabel('Time hr'); ylabel('Concentration #'); legend('Plasma','Muscle', 'Fat','Liver','Other Peri');

% figure LDL-III; subplot(2,2,3) hold on plot(t,C(:,29),... 'g') plot(t,C(:,30),... 'm') plot(t,C(:,31),... 'b') plot(t,C(:,32),... 'r') plot(t,C(:,33),... 'k') % plot(tI,CI(:,29),... % 'g') % plot(tI,CI(:,30),... % 'm') % plot(tI,CI(:,31),... % 'b') % plot(tI,CI(:,32),... % 'r') % plot(tI,CI(:,33),... % 'k') title('LDL-III'); xlabel('Time hr'); ylabel('Concentration #'); legend('Plasma','Muscle', 'Fat','Liver','Other Peri');

%figure PREHDL; subplot(2,2,1) hold on plot(t,C(:,39),... 'g') plot(t,C(:,40),... 'm') plot(t,C(:,41),... 'b') plot(t,C(:,42),... 'r') plot(t,C(:,43),... 'k') % plot(tI,CI(:,39),... % 'g') % plot(tI,CI(:,40),... % 'm')

% plot(tI,CI(:,41),... % 'b') % plot(tI,CI(:,42),... % 'r') % plot(tI,CI(:,43),... % 'k') title('PREHDL'); xlabel('Time hr'); ylabel('Concentration #'); legend('Plasma','Muscle', 'Fat','Liver','Other Peri');

%figure HDL-I; subplot(2,2,2) hold on plot(t,C(:,44),... 'g') plot(t,C(:,45),... 'm') plot(t,C(:,46),... 'b') plot(t,C(:,47),... 'r') plot(t,C(:,48),... 'k') % plot(tI,CI(:,44),... % 'g') % plot(tI,CI(:,45),... % 'm') % plot(tI,CI(:,46),... % 'b') % plot(tI,CI(:,47),... % 'r') % plot(tI,CI(:,48),... % 'k') title('HDL-I'); xlabel('Time hr'); ylabel('Concentration #'); legend('Plasma','Muscle', 'Fat','Liver','Other Peri');

%figure HDL-II; subplot(2,2,3) hold on plot(t,C(:,49),... 'g') plot(t,C(:,50),... 'm') plot(t,C(:,51),... 'b') plot(t,C(:,52),... 'r') plot(t,C(:,53),... 'k') % plot(tI,CI(:,49),... % 'g')

% plot(tI,CI(:,50),... % 'm') % plot(tI,CI(:,51),... % 'b') % plot(tI,CI(:,52),... % 'r') % plot(tI,CI(:,53),... % 'k') title('HDL-II'); xlabel('Time hr'); ylabel('Concentration #'); legend('Plasma','Muscle', 'Fat','Liver','Other Peri');

%figure HDL-C; subplot(2,2,4) hold on plot(t,C(:,49)*HDLIC+C(:,49)*HDLC ,... 'g') % plot(tI,CI(:,49)*HDLIC+CI(:,49)*HDLC ,... % 'g') title('HDL-C'); xlabel('Time hr'); ylabel('Concentration mol'); legend('Plasma');

%figure Cholesterol; subplot(2,2,1) hold on plot(t,C(:,35),... 'm') plot(t,C(:,36),... 'b') plot(t,C(:,37),... 'r') plot(t,C(:,38),... 'k') % plot(tI,CI(:,35),... % 'm') % plot(tI,CI(:,36),... % 'b') % plot(tI,CI(:,37),... % 'r') % plot(tI,CI(:,38),... % 'k') title('Cholesterol'); xlabel('Time hr'); ylabel('Concentration Mol/L'); legend('Muscle', 'Fat','Liver','Other Peri');

%figure TG;

73 subplot(2,2,2) hold on plot(t,C(:,54),... 'g') plot(t,C(:,55),... 'm') plot(t,C(:,56),... 'b') plot(t,C(:,57),... 'r') % plot(tI,CI(:,54),... % 'g') % plot(tI,CI(:,55),... % 'm') % plot(tI,CI(:,56),... % 'b') % plot(tI,CI(:,57),... % 'r') title('Triglyceride'); xlabel('Time hr'); ylabel('Concentration Mol/L'); legend('Muscle', 'Fat','Liver','Other Peri');

%figure FA; subplot(2,2,3) hold on plot(t,C(:,58),... 'g','LineWidth',2) plot(t,C(:,59),... 'm') plot(t,C(:,60),... 'b') plot(t,C(:,61),... 'r') plot(t,C(:,62),... 'k') % plot(tI,CI(:,58),... % 'g','LineWidth',2) % plot(tI,CI(:,59),... % 'm') % plot(tI,CI(:,60),... % 'b') % plot(tI,CI(:,61),... % 'r') % plot(tI,CI(:,62),... % 'k') title('Fatty Aicd'); xlabel('Time hr'); ylabel('Concentration Mol/L'); legend('Plasma','Muscle', 'Fat','Liver','Other Peri');

%figure VLDLC; subplot(2,2,4) hold on plot(t,C(:,64)) % plot(tI,CI(:,64)) title('Cholesterol of VLDL') xlabel('Time hr'); ylabel('mol per particle'); legend('bolus'); %legend('infusion');

%mol/L * L *

%figure TG mass;

% famo = 931.42; % hold on % % plot(tI,CI(:,54)*V3*famo,... % 'g') % plot(tI,CI(:,55)*V4*famo,... % 'm') % plot(tI,CI(:,56)*V2*famo,... % 'b') % plot(tI,CI(:,57)*V5*famo,... % 'r') % title('Triglyceride'); % xlabel('Time hr'); % ylabel('mass g per tissue'); % legend('Muscle', 'Fat','Liver','Other Peri');

hold on plot(tI,CI(:,56),... 'b') title('Liver Triglyceride'); xlabel('Time hr'); ylabel('Concentration Mol/L'); legend('Triglyceride'); plot(tI,CI(:,37),... 'r') title('Liver Cholesterol'); xlabel('Time hr'); ylabel('Concentration Mol/L'); legend('Cholesterol');

Appendix III Matlab Code for Modified Gillespie algorithms

% runs algorithm without input states from previous run function [t_span, orgs, LDL] = FAtransport_6init() close all;

% matrix parameters numVLDL = 69; numIDL = 32; numLDL= 1272; time_step = 0.1; % hr/4 t_max = 50; % hr/4 t_span = 0:time_step:t_max;

% initial lipid composition frac_sat = 1; frac_MU = 0; frac_PU = 0; frac_trans = 0;

% Initialize matrix % 1 sat TG, 2 micelle type (0 - LDL, 1 - HDL, 2 - chylomicron) % 3 PU TG (unused), 4 trans TG (unused) % 5 sat CE, 6 MU CE (unused), 7 PU CE (unused), 8 trans CE (unused) % 9 oxidation level, 10 organ % 11 ApoC, 12 time in organ, 13 num HL, 14 HL hydrolyzed, % 15 CETP present micdim = 15; lset = zeros(numLDL+numIDL+numVLDL, micdim);

mu_TG = 5200; mu_CE = 3100; CM_TG = 6180000/16; CM_CE = 611000/16; HDL_TG = 20; HDL_CE = 20;

VmaxHL = 0.7; % 1/h KmHL = 1000;

t_liver = 19.2/3600; % h t_musc = 62.5/3600; t_adip = 62.5/3600; t_int = 62.5/3600; t_periph = 62.5/3600; t_plasma = 60/3600;

f_HL = 0.01;

% % initial VLDL % lset(1:numVLDL,1) = normrnd(frac_sat*mu_TG, 500, [numVLDL, 1]); % lset(1:numVLDL,5) = normrnd(frac_sat*mu_CE, 500, [numVLDL, 1]); % lset(1:numVLDL,11) = 10; % % % initial IDL % lset(numVLDL+1:numVLDL+numIDL,1) = normrnd(frac_sat*mu_TG/10, 50, [numIDL, 1]); % lset(numVLDL+1:numVLDL+numIDL,5) = normrnd(frac_sat*mu_CE/10, 50, [numIDL, 1]); % % % initial LDL % lset(numVLDL+numIDL+1:numVLDL+numIDL+numLDL,1) = normrnd(frac_sat*mu_TG/100, 50, [numLDL, 1]); % lset(numVLDL+numIDL+1:numVLDL+numIDL+numLDL,5) = normrnd(frac_sat*mu_CE/100, 50, [numLDL, 1]); % % % initial HDL % lset(1:1000,1) = normrnd(frac_sat*HDL_TG, HDL_TG/10, [numLDL, 1]); % lset(1:1000,5) = normrnd(frac_sat*HDL_CE, HDL_CE/10, [numLDL, 1]);

LDL = lset;

liverFA = 1e5; plasmaFA = 2e5; muscleFA = 2e5; adipFA = 4e6; intFA = 0; periphFA = 0;

% breakfast meal = 100000000/16; intFA = intFA + meal

intFAmax = intFA; liverFAmax = intFA;

CETP_TG = 18.6; CETP_CE = 18.6;

% loop through time steps for i = 1:length(t_span)

% % meals (lunch & dinner) % if i*time_step == 20 || i*time_step == 40 % intFA = intFA + meal; % end %

orgs(:,i) = [plasmaFA, liverFA, muscleFA, adipFA, intFA, periphFA];

state_curr = LDL(:,:,length(LDL(1,1,:))); state_next = zeros(size(state_curr));

% revise conc_apoA = 4;

% add new micelles mic2add = zeros(1, length(state_curr(1, :)));

% generate CM from intestine if intFA > 0

% calculate potential CM mu_released = intFA/(3*CM_TG + 1*CM_CE);

% number to release (fraction of total possible) num_released = round(normrnd(mu_released, mu_released/10));

% form CM if num_released > 0 newCM = zeros(num_released, micdim); newCM(:,2) = 2; % CM tag newCM(:,1) = normrnd(frac_sat*CM_TG, 500, [num_released,1]); % satTG newCM(:,5) = normrnd(frac_sat*CM_CE, 500, [num_released,1]); % satCE newCM(:,10) = 4; % organ (intestine) newCM(:,11) = 20; % ApoCII newCM(:,12) = normrnd(t_int, t_int/10, [num_released,1]); % time in organ

while 3*sum(newCM(:,1))+sum(newCM(:,5)) > intFA newCM = newCM(1:length(newCM(:,1))-1,:); end

mic2add = [mic2add; newCM]; intFA = intFA - 3*sum(newCM(:,1)) - sum(newCM(:,5));

% disp(i); % disp(num_released); end end

% generate VLDL from liver if liverFA > 0

% calculate potential VLDL mu_released = liverFA/(3*mu_TG + 1*mu_CE)/15;

% number to release (fraction of total possible) num_released = round(normrnd(mu_released, mu_released/10));

% release VLDL if num_released > 0 newVLDL = zeros(num_released, micdim); newVLDL(:,2) = 0; % VLDL tag newVLDL(:,1) = normrnd(frac_sat*mu_TG, 500, [num_released,1]); % satTG newVLDL(:,5) = normrnd(frac_sat*mu_CE, 500, [num_released,1]); % satCE newVLDL(:,10) = 1; % organ (liver) newVLDL(:,11) = 10; % ApoCII newVLDL(:,12) = normrnd(t_liver, t_liver/10, [num_released,1]); % time in organ

while 3*sum(newVLDL(:,1))+sum(newVLDL(:,5)) > intFA newVLDL = newVLDL(1:length(newVLDL(:,1))-1,:); end

mic2add = [mic2add; newVLDL]; liverFA = liverFA - 3*sum(newVLDL(:,1)) - sum(newVLDL(:,5)); end

end

% generate HDL from plasma if plasmaFA > 0

liverFA = liverFA + 0.03*plasmaFA; plasmaFA = 0.97*plasmaFA; % % calculate potential VLDL % mu_released = plasmaFA/(3*HDL_TG + 1*HDL_CE); % % % number to release (fraction of total possible) % num_released = round(normrnd(mu_released, mu_released/10)); % % % release HDL % if num_released > 0 % newHDL = zeros(num_released, micdim); % newHDL(:,2) = 1; % HDL tag % newHDL(:,1) = normrnd(frac_sat*HDL_TG, HDL_TG/10, [num_released,1]); % satTG % newHDL(:,5) = normrnd(frac_sat*HDL_CE, HDL_CE/10, [num_released,1]); % satCE % newHDL(:,10) = 0; % organ (plasma) % newHDL(:,12) = normrnd(t_liver, t_liver/10, [num_released,1]); % time in organ % % mic2add = [mic2add; newHDL]; % plasmaFA = liverFA - 3*sum(newHDL(:,1)) - sum(newHDL(:,5)); % end end

% basal metabolic rate BMR = meal*time_step/24;

if muscleFA > BMR*.85 muscleFA = muscleFA - BMR*.85; else

muscleFA = 0; end

% adipose rerelease if adipFA > BMR/2 adipFA = adipFA - BMR/2; plasmaFA = plasmaFA + BMR/2; else plasmaFA = adipFA; adipFA = 0; end % plasmaFA = plasmaFA+0.03*adipFA; % adipFA=0.97*adipFA;

% liver absorption through plasma liverFA = liverFA+0.03*plasmaFA; plasmaFA = 0.97*plasmaFA;

% loop through micelles for j = 1:length(state_curr(:,1))

LDL_curr = state_curr(j, :);

% check if micelle still exists if LDL_curr(1) > 0

% Organ values: 0 - plasma, 1 - liver, 2 - muscle, 3 - adipose, % 4 - intestine, 5 - other peripheral

% distribution if LDL_curr(12) > 0 LDL_curr(12) = LDL_curr(12) - time_step;

elseif LDL_curr(10) ~=0 LDL_curr(10) = 0; LDL_curr(12) = normrnd(t_plasma, t_plasma/10);

elseif LDL_curr(10) == 0

k_plasmatoliver = 56; k_plasmatomusc = 56; k_plasmatoadip = 56; k_plasmatoint = 56; k_plasmatoperiph = 56; k_stay = 56;

k = [k_plasmatoliver k_plasmatomusc k_plasmatoadip k_plasmatoint k_plasmatoperiph k_stay]; k_dist = sum(k); r_dist = rand;

% plasma -> liver if r_dist < k(1)/k_dist LDL_curr(10) = 1; LDL_curr(12) = normrnd(t_liver, t_liver/10);

% plasma -> muscle elseif r_dist < sum(k(1:2))/k_dist LDL_curr(10) = 2; LDL_curr(12) = normrnd(t_musc, t_musc/10);

% plasma -> adipose elseif r_dist < sum(k(1:3))/k_dist LDL_curr(10) = 3; LDL_curr(12) = normrnd(t_adip, t_adip/10);

% plasma -> intestine elseif r_dist < sum(k(1:4))/k_dist LDL_curr(10) = 4; LDL_curr(12) = normrnd(t_int, t_int/10);

% plasma -> peripheral tissue elseif r_dist < sum(k(1:5))/k_dist LDL_curr(10) = 5; LDL_curr(12) = normrnd(t_periph, t_periph/10); else LDL_curr(12) = normrnd(t_plasma, t_plasma/10); end

end

% HL reaction (Michaelis-Menten) if LDL_curr(13) > 0 && LDL_curr(1) > 37 % LDL_curr(14) = (LDL_curr(13)*f_HL+(1-LDL_curr(13)*f_HL)*LDL_curr(14)); LDL_curr(14) = LDL_curr(14)+ LDL_curr(13)*VmaxHL*LDL_curr(1)/(KmHL+LDL_curr(1)); end

% find rates k = calc_rates(LDL_curr, state_curr, conc_apoA, orgs);

% determine rest time and whether to change state k_tot = sum(k); rest_time = -1/(k_tot)*log(1-rand);

% determine next state if rest_time

r = rand;

% LPL if r < k(1)/k_tot

LDL_curr(1) = LDL_curr(1)*.790; LDL_curr(5) = LDL_curr(5)*.869; LDL_curr(11) = LDL_curr(11)-1;

% add into appriopriate tissue if LDL_curr(10) == 2 muscleFA = muscleFA + 3*(LDL_curr(1)/.790-LDL_curr(1)) + ... (LDL_curr(5)/.869-LDL_curr(2)); else adipFA = adipFA + 3*(LDL_curr(1)/.790-LDL_curr(1)) + ... (LDL_curr(5)/.869-LDL_curr(2)); end

% LDL -> LDL+HL elseif r < sum(k(1:2))/k_tot LDL_curr(13) = LDL_curr(13)+1;

% LDL+HL -> hydrolysis elseif r < sum(k(1:3))/k_tot

LDL_curr(1) = LDL_curr(1)-LDL_curr(14); LDL_curr(5) = LDL_curr(5)-LDL_curr(14);

if LDL_curr(14) > HDL_TG % spawn HDL newHDL = zeros(1, length(LDL_curr)); newHDL(2) = 1; % HDL tag newHDL(1) = round(normrnd(frac_sat*HDL_TG, HDL_TG/10)); % satTG newHDL(5) = round(normrnd(frac_sat*HDL_CE, HDL_CE/10)); % satCE newHDL(10) = LDL_curr(10); % organ (same as LDL) newHDL(12) = LDL_curr(12); % time in organ

mic2add = [mic2add; newHDL];

% add to plasma plasmaFA = plasmaFA + 3*(LDL_curr(14)-newHDL(1)) + LDL_curr(14)-newHDL(5);

else plasmaFA = plasmaFA + 3*LDL_curr(14) + LDL_curr(14); end

% remove HL LDL_curr(13) = LDL_curr(13)-1; LDL_curr(14) = 0;

% LDL receptor elseif r < sum(k(1:4))/k_tot % liver metabolism/storage mictype = LDL_curr(2);

if LDL_curr(10) == 1 liverFA = liverFA + 3*LDL_curr(1) + LDL_curr(5); elseif LDL_curr(10) == 2 muscleFA = muscleFA + 3*LDL_curr(1) + LDL_curr(5); elseif LDL_curr(10) == 3 adipFA = adipFA + 3*LDL_curr(1) + LDL_curr(5); end

LDL_curr = zeros(length(LDL_curr),1); LDL_curr(2) = mictype;

% CETP first attachment elseif r < sum(k(1:5))/k_tot LDL_curr(15) = 1;

% CETP second attachment elseif r < sum(k(1:6))/k_tot org = LDL_curr(10);

% choose other micelle mic_poss = find((state_curr(:,2) == 0 | state_curr(:,2) == 1)... & state_curr(:,10) == org & state_curr(:,15) == 0);

otherind = datasample(mic_poss,1);

if otherind < j LDL_other = state_next(otherind,:); else LDL_other = state_curr(otherind,:); end

% transfer lipids if LDL_curr(2) ~= LDL_other(2) || LDL_curr(1)/LDL_other(1)>5 ||... LDL_curr(1)/LDL_other(1)<0.2 if LDL_curr(1)>LDL_other(1) LDL_curr(1) = LDL_curr(1) - CETP_TG; LDL_other(1) = LDL_other(1) + CETP_TG; LDL_curr(5) = LDL_curr(5) + CETP_CE; LDL_other(5) = LDL_other(5) - CETP_CE; else LDL_curr(1) = LDL_curr(1) + CETP_TG; LDL_other(1) = LDL_other(1) - CETP_TG; LDL_curr(5) = LDL_curr(5) - CETP_CE; LDL_other(5) = LDL_other(5) + CETP_CE; end end

LDL_curr(15) = 0;

% replace LDL_other

if otherind < j state_next(otherind,:) = LDL_other; else state_curr(otherind,:) = LDL_other; end end

end

% fill next state state_next(j,:) = LDL_curr; else state_next(j,:) = LDL_curr; end end

% add new formed micelles

for k = 2:length(mic2add(:,1)) type = mic2add(k,2); ind = find(state_next(:,1) == 0 & state_next(:,2) == type);

if ~isempty(ind) state_next(ind(1),:) = mic2add(k,:); else state_next = [state_next; mic2add(k,:)]; end end

new_mic = length(state_next(:,1)) - length(LDL(:,1,1));

if length(state_next(:,1)) > length(LDL(:,1,1)) LDL = cat(1, LDL, zeros(new_mic, length(LDL(1,:,1)), length(LDL(1,1,:)))); end

% store next state LDL = cat(3, LDL, state_next);

disp(strcat(num2str(i/length(t_span)*100), '%'));

end

function k = calc_rates(LDL_curr, state_curr, conc_apoA, orgs)

muscleFA = orgs(3);

% LPL k(1) if LDL_curr(11) > 0 && LDL_curr(2) ~= 1 && (LDL_curr(10) == 2 || LDL_curr(10) == 3) k_LPL = 10; else k_LPL = 0; end k = k_LPL;

% LDL -> LDL+HL k(2) if LDL_curr(1) > 37 && LDL_curr(2) == 0 && LDL_curr(1)<800 k_HL = 800; else k_HL = 0; end k = [k k_HL];

% LDL+HL -> hydrolysis k(3) if LDL_curr(13)>0 && LDL_curr(2) == 0 if LDL_curr(1) > 37 && LDL_curr(14)

% LDL receptor k(4) kmet_liver = 5/4; %0.1; kmet_musc = 0.5/3;%*(0.5e5/muscleFA); %0.1; kmet_adip = 0.5/16; %0.1; kmet_int = 0; %0.1; kmet_periph = 0; %0.1; if LDL_curr(2) ~= 1

if LDL_curr(10) == 1 && LDL_curr(1) < 1e6 k = [k kmet_liver]; elseif LDL_curr(10) == 2 && LDL_curr(2) == 0 k = [k kmet_musc]; elseif LDL_curr(10) == 3 && LDL_curr(2) == 0 k = [k kmet_adip]; elseif LDL_curr(10) == 4 k = [k kmet_int]; elseif LDL_curr(10) == 5 k = [k kmet_periph]; else k = [k 0]; end

elseif LDL_curr(10) == 1 && LDL_curr(10) == 1 k = [k kmet_liver*100]; else k = [k 0]; end

% CETP first attachment k(5) k_CETP1 = 1;

if LDL_curr(15) == 0 k = [k k_CETP1]; else k = [k 0]; end

% CETP second attachment k(6) if LDL_curr(15) == 1 org = LDL_curr(10); num_poss = size(find((state_curr(:,2) == 0 | state_curr(:,2) == 1) ... & state_curr(:,10) == org & state_curr(:,15) == 0)); k_CETP2 = num_poss*0.001; else k_CETP2 = 0; end

k = [k k_CETP2]; end

% plots model outputs function noout = timeplotter(t_span, orgs, mic)

% [t_span, orgs, mic] = FAtransport_6();

t_span = t_span./4;

indLDL = find(mic(:,2,length(mic(1,1,:))) == 0); indHDL = find(mic(:,2,length(mic(1,1,:))) == 1 & mic(:,1,length(mic(1,1,:))) ~= 0); indCM = find(mic(:,2,length(mic(1,1,:))) == 2);

lowD = mic(indLDL,:,:); HDL = mic(indHDL,:,:); chylo = mic(indCM,:,:);

% 1 VLDL, 2 IDL, 3 LDL, 4 HDL, 5 CM, 6 CMR TG = zeros(length(t_span),6); CE = zeros(length(t_span),6); nmic = zeros(length(t_span),6);

for i = 1:length(t_span); VLDL = lowD(find(lowD(:,1,i) > 800),:,i); IDL = lowD(find(lowD(:,1,i) <= 800 & lowD(:,1,i) >= 200),:,i); LDL = lowD(find(lowD(:,1,i) < 200 & lowD(:,1,i) > 0),:,i); CM = chylo(find(chylo(:,1,i) > 800), :, i); CMR = chylo(find(chylo(:,1,i) <= 800 & chylo(:,1,i) > 0), :, i);

% triglyceride per species TG(i,1) = mean(VLDL(:,1)); TG(i,2) = mean(IDL(:,1)); TG(i,3) = mean(LDL(:,1)); TG(i,4) = mean(HDL(:,1,i)); TG(i,5) = mean(CM(:,1)); TG(i,6) = mean(CMR(:,1));

% cholesterol ester per species CE(i,1) = mean(VLDL(:,5)); CE(i,2) = mean(IDL(:,5)); CE(i,3) = mean(LDL(:,5)); CE(i,4) = mean(HDL(:,5,i)); CE(i,5) = mean(CM(:,5)); CE(i,6) = mean(CMR(:,5));

% amount of micelles per species nmic(i,1) = length(VLDL(:,1)); nmic(i,2) = length(IDL(:,1)); nmic(i,3) = length(LDL(:,1)); nmic(i,4) = length(HDL(:,1,i)); nmic(i,5) = length(CM(:,1)); nmic(i,6) = length(CMR(:,1)); end

% figure; % hold on; % % plot(t_span, TG(:,1)); % plot(t_span, TG(:,2)); % plot(t_span, TG(:,3)); % % plot(t_span, TG(:,4)); % % plot(t_span, TG(:,5)); % % plot(t_span, TG(:,6)); % % legend('VLDL', 'IDL', 'LDL', 'HDL', 'CM', 'CMR'); % xlabel('Time (hr)'); % ylabel('Micelle TG content (molecules)'); % title('Micelle TG Content'); %

% hold off;

% lipoprotein body concentration figure; hold on;

% plot(t_span, nmic(:,1)); % plot(t_span, nmic(:,2)); % plot(t_span, nmic(:,3)); % plot(t_span, nmic(:,5)); % plot(t_span, nmic(:,6)); plot(t_span, nmic(:,4));

% legend('VLDL', 'IDL', 'LDL', 'CM', 'CMR', 'HDL'); legend('HDL'); xlabel('Time (hr)'); ylabel('Amount micelles (count)'); title('Amount Micelles');

hold off;

% organ composition figure;

hold on; plot(t_span, orgs(1,:)); plot(t_span, orgs(2,:)); plot(t_span, orgs(3,:)); % plot(t_span, orgs(4,:)); % plot(t_span, orgs(5,:)); % plot(t_span, orgs(6,:));

legend('plasma', 'liver', 'muscle', 'adipose', 'intestines', 'peripheral'); xlabel('Time (hr)'); ylabel('Organ lipid amount (molecules)'); title('Organ lipid amount');

hold off;

% Lipoprotein TG figure;

hold on; % plot(t_span, TG(:,1)); % plot(t_span, TG(:,2)); % plot(t_span, TG(:,3)); % plot(t_span, TG(:,4)); plot(t_span, TG(:,5)); % plot(t_span, TG(:,6));

% legend('VLDL', 'IDL', 'LDL', 'CM', 'CMR', 'HDL'); legend('CM', 'CMR', 'HDL'); xlabel('Time (hr)'); ylabel('Micelle TG composition (molecules/micelle)'); title('Micelle TG composition');

hold off;

% lipoprotein CE figure;

hold on; % plot(t_span, CE(:,1)); % plot(t_span, CE(:,2)); % plot(t_span, CE(:,3)); % plot(t_span, CE(:,4)); plot(t_span, CE(:,5)); % plot(t_span, CE(:,6));

% legend('VLDL', 'IDL', 'LDL', 'CM', 'CMR', 'HDL'); legend('CM', 'CMR', 'HDL'); xlabel('Time (hr)'); ylabel('Micelle CE composition (molecules/micelle)'); title('Micelle CE composition');

hold off; end

% runs chosen simulation and plots

% [t_span, orgs, mic]=FAtransport_6init(); % [t_span, orgs, mic]=FAtransport_6(micinit, orgsinit); close all; timeplotter(t_span, orgs, mic);

% saves final values to initialize with

89 micinit = mic(:,:,length(mic(1,1,:))); orgsinit = orgs(:,length(orgs(1,:)));

Curriculum Vitae Minxue Jia

EDUCATION: Johns Hopkins University (JHU), Baltimore, MD August 2016 - May 2018 MS, Chemical and Biomolecular Engineering

Worcester Polytechnic Institute (WPI), Worcester, MA August 2012 - May 2016 BS, Double Major in Chemistry & Chemical Engineering (biological concentrating)

RESEARCH EXPERIENCE：

Cholesterol and Heart Disease Simulation Project, Donohue ChemBE Lab, JHU December 2016 - May 2018

 Built physiologically based pharmacokinetic ODE model and Gillespie algorithm through Matlab and Mathematica  Simulated the influential effect of dietary lipid for lipoprotein transport and lipid metabolism Nano-Bioengineering Lab Project, JHU January 2017 - May 2017

 Worked on a team of three to investigate microfluidic device and enhance the polydispersity of cationic polymer gene nanoparticles  Designed a novel microfluidic device, performed wet-lab experiment and tuned finely the mixing strategy  Examined influential effect of device shape, flow rate for mixing efficiency through performing COMSOL simulation Major Qualifying Project (MQP), Zoll Cellars, Shrewsbury, MA August 2015 - May 2016

 Improved the quality of red wine for Zoll Cellars through investigating influential fermentation factors for grape  Optimized fermentation process by evaluating PH, sugar content and the yeast assimilable nitrogen (YAN)  Qualified the wine product and used GC-MS & NMR to analyze aromatic compound originated from wine Undergraduate Researcher, Goddard Deskins Chemical Engineering Lab, WPI October 2015 - May 2016

 Conducted investigation into vibrational characterization of hydrothermal chars through WebMo simulation  Investigated the catalysis mechanism of metal ions for decomposition of polystyrenes Undergraduate Researcher, Gateway Park Emmert Chemistry Lab, WPI March 2014 - March 2015

 Developed electrophilic amination reagents for Copper Catalyzed Non-directed Benzylic C-H Amination  Evaluated influential factors of catalytic reaction and determined product yields through NMR spectroscopy TEACHING EXPERIENCE：

Teaching Assistant, Chemical and Biomolecular Engineering Department, JHU January 2017 - May 2018

 Assisted students with performing laboratory experiments for AlChE Chem Car project among a class of 19 students  Determined specific lab safety and articulated procedures to be followed Teaching Assistant, Chemical Engineering Department, WPI August 2015 - May 2016

 Held weekly office hours to answer students’ questions, evaluated students’ assignments and monitored exams (Heat Transfer, Mass Transfer, Fluid Mechanics, Reaction Design) SCHOLARSHIP & REWARDS:

 ChemBE Master’s Essay Scholarship, JHU August 2017 - May 2018  Member, Tau Beta Pi (Engineering) Honor Society April 2015 - Present  University Award, WPI August 2012 - May 2016

LEADERSHIP EXPERIENCE:

 Public Relations Chair, Chinese Student Association February 2015 - May 2016  Dumpling Program Chair, Chinese Student Association October 2014 - May 2016