Alex Rockhill and Gerard Trimberger AMATH 423 March 14, 2016

Modeling Energy Metabolism

A red blood cell seen under an electron microscope. Red blood cells carry oxygen and carbon dioxide, the fuel and byproduct respectively of aerobic respiration, the process that underlies our success as a species (Pretorius, 2013).

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Abstract

The three metabolic systems that create energy in our body, phosphocreatine (PC),

anaerobic respiration and aerobic respiration, were modeled using assumptions about the

interactions of the components of these systems. Using the core principles of physics and

relevant observations of biological organisms, we were able make logical assumptions that we

could use to model energy output of these three systems. The high dimensionality of the

system prevented exact analytical solutions, but by using separation of time scales and

Bayesian inference to fit the model to data, the relationships between variables and initial

conditions were able to be studied. These initial conditions and variables which yielded the

best fits were analyzed and applied to current problems in science, medicine and lifestyle.

These models allowed us to examine the way our body generates energy and the conditions

necessary to improve this process.

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Introduction

Many people may not consider the concept of chemical potential on a daily basis, but it

is essential to the way we obtain energy. All of our cells need energy for homeostatic

maintenance and to perform their function. This energy is in the form of high energy molecules

such as adenosine triphosphate (ATP). These molecules have energy because of their chemical

potential; they are in a stable arrangement but can be rearranged by a chemical reaction to be

in a more stable state. By themselves two atoms each have their own energies, but the total of

the combination of these two energies can be decreased when these two atoms are interact

with each other in the form of a chemical bond. In essence, we observe that oppositely

charged particles attract, and so it takes energy to separate the positive and negatively

charged particle. Between the nucleus and electrons of an atom, this is called the weak force.

The energy due to the separation of charges in a lone atom can be decreased by bringing

another positively charged nucleus close enough so that the electrons of both nuclei can be

close to two opposite charges instead of one. Logically, some configurations of nuclei and

electrons would lead to more even sharing of electrons and therefore lower energy and some

more uneven sharing and therefore higher energy. By taking in molecules with large amounts of

chemical potential energy, those that have bonds with electrons unevenly shared, we are able

to use biological processes to convert these molecules into other molecules with less energy,

where chemical bonds share electrons more evenly (Solomons et al. 2013). This energy can

then be converted into forms of energy that may be more familiar to many people such as work

and heat through other biological processes.

The chemical reactions underlying these biological processes that create ATP have

been well described. There are three systems of chemical reactions create ATP that our body

uses as energy: phosphocreatine (PC), anaerobic (Gan) and aerobic (Gaero) respiration. Our body

maintains stores of phosphocreatine in our muscles that can be converted directly into ATP via

the following reaction:

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(1)ATP + Cr ⌧ PCr +ADP +H+

This reaction creates energy the fastest and is used for short intense energy use (Clark, 1997).

Anaerobic respiration uses lactic acid fermentation to convert glucose into energy,

which does not require oxygen. The chemical reaction for this process is:

The time scale is slower, causing energy use to be more distributed over time. The anaerobic

energy system is therefore used for energy use that is less intense than the PC system. In order

for this reaction to occur, glucose must go through glycolysis and then the citric acid cycle.

The third system of chemical reactions that creates ATP is aerobic respiration. The net

reaction is as follows:

In this system, fats and glucose are broken down into pyruvate which is then used in the Kreb’s

Cycle. This system is the most efficient and occurs on the slowest time scale, contributing the

most energy for over longer periods of the least intense energy usage (Freeman, 2010).

Together, these three energy systems enable us to live and function as people. Energy

for thinking, talking, moving, breathing and everything else that defines what it means to be

alive as a human is provided by these energy systems. Much of this energy is provided

aerobically to fuel the thoughts that have made us so successful as a species and the basic

processes that allow our bodies to function. However, in periods of moderate to intense

exercise, all three energy systems are involved. These energy systems are observed to

complement each other; when one system is at a peak, the others are preparing for energy use

or already spent. By studying the process of metabolism at periods of energy usage that span

all three energy systems with mathematical models, we hope to learn about how these systems

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(2)Glucose+ 2ADP + 2Pi ⌧ 2ATP + 2H2O + 2LA

(3)Glucose+ 6O2 + 32ADP + 32Pi ⌧ 32ATP + 6H2O + 6CO2

function and interact, and how to best utilize these systems and their complementarity. This

model will likely be applicable to peak athletic performance, but may also be applicable to all

sorts of dysfunction of the energy system, from obesity to stroke.

Problem formulation and Models

When it comes to creating a generalized model for a system as complicated as human

metabolism, it becomes quite evident that many simplifying assumptions will need to be made

to create a system that can even be solved. In 2013, a paper by Thiele et. al. presented Recon

2, a ‘community-driven global reconstruction of human metabolism.’ The focus of this paper

was to create a ‘Google Maps’ of Human Metabolism so it was their goal to create “the most

comprehensive representation of the human metabolism that is applicable to computational

modeling.” The search was based on the idea that many models for human metabolism have

been constructed; however each of them only represents a subset of our knowledge (Thiele,

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Performance and Metabolism of World Record Runners

Distance (m) Time (Men) Time (Women) Approximate % VO2 Max

% Phosphate

% Lactate

% Aerobic

Velocity

100 10s 11s N/A 70 22 8 10.2

200 20s 22s N/A 40 46 14 10.1

400 43s 48s N/A 10 60 30 9.2

800 102s 113s 135 5 38 57 7.9

1,500 209s 232s 112 2 22 76 7.2

3,000 449s 503s 102 <1 12 88 6.7

5,000 778s 877s 97 <1 7 93 6.4

10,000 1628s 1814s 92 <1 3 97 6.1

42,195 7610s 8466s 82 <1 <1 99 5.5

Table 1. World Record times (in seconds) for both men and women are shown for distances (in meters) from the 100m to the marathon. The contributions of each energy system to the total energy expended are tabulated for each event (Martin and Coe, 1991). Velocities were calculated based on distance over time.

2013). Data driven models such as this can have provide insight, but take much more time and

effort than mechanistic models which may be just as effective if done well.

Using the chemical reactions that govern the three systems of energy, we formulated a

system of differential equations based on mechanistic principles. These models rely on

assumptions based on these chemical reactions and physics. Our model was formulated using

the conservation of energy (E); we modeled that all energy expended in the form effort/pace

(output) is equal and opposite to energy generated (input).

We modeled energy as being proportional to ATP under the assumption that ATP provides all

the energy. While this assumption is not completely accurate, guanine triphosphate (GTP) and

other high energy molecules can be used as forms of energy, ATP is the predominant molecule

providing energy (Freeman, 2012). We also initially assumed that there are no ATP stores; it is

used as soon as it is made in the time scale of our model. ATP is a highly energetic molecule

(Solomons, 2013) and so from one second to the next all of the ATP created has been used.

Therefore, we will model the rate of change of metabolites and measure the output of the

change in ATP. Since PC generates ATP in direct proportion, anaerobic respiration generates

two molecules of ATP for every glucose and aerobic respiration ideally generates 32 molecules

of ATP for every molecule of glucose, we modeled the change in ATP as:

Aerobic respiration creates 32 ATP ideally, using glucose, but often the yield is not as great

because this ATP must be used to transport molecules necessary for this process (Freeman,

2012). Fat was also modeled, as being involved in aerobic respiration but having a much lower

yield due to transportation and energy cost from being broken down into free fatty acids and

subsequently acetyl coenzyme which is the precursor for the Kreb’s Cycle (Solomons, 2013).

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dE

dt

= input� output

dATP

dt

= PC

out

+ 32 or2Gout

+ F

out

While both of these measurements are inexact, the their relationship to the other variables is

likely qualitatively correct.

The differential equation for each metabolite was then modeled by input and output

terms for production and use respectively. Additionally, we assumed that the body would use

each metabolite at its particular max rate when that metabolite is in excess but below if there is

not enough metabolite to use. This assumptions is justified by the physiological limits of the

body’s energy system— even if there is an excess of a metabolite, there still have to be

enzymes or mitochondria to process these into ATP (Solomons, 2013). Logically, if there are

few metabolite molecules, the body cannot use this energy system as its main source. The

production term was modeled according to assumptions for negative feedback loops, which

are ubiquitous in biological systems (Freeman, 2012). If there is more of a metabolite, that

metabolite will inhibit the body’s production of more of itself. If that metabolite is at low

concentration, its inhibition on its own synthesis will be weak and the body will produce more

of it. Based on these principles, we formed the following differential equations:

The oxygen and lactic acid terms in the above equation are derived from similar assumptions.

Since oxygen is required in aerobic respiration, when oxygen is in excess, we assumed that it

too could only be burned at a maximum rate limited by the physiology of the body. At

sufficiently low concentrations oxygen would inhibit aerobic respiration. Lactic acid was

assumed to have no effect on the anaerobic reaction at sufficiently low concentrations, but as

its concentration increased it would inhibit the reaction. Aerobic respiration using fat was

modeled as depending on oxygen but not fat concentrations because these were assumed to

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(4)dPC

dt= �r(

PC

1 + PC) + p(PC

max

� PC)

(5)G

aero

dt= �c

O2

1 +O2

G

1 +G

(6)dG

an

dt= �b

G

1 +G

1

1 + LA

(7)dF

dt= �d

O2

1 +O2

b, c, d, r, p > 0

be in excess. On the time scale of our model, we assumed that the fat concentration in a

typical human body was sufficient it would not be a limiting factor in the burning of fat.

Although there is more complexity to mobilizing of fat as an energy resource, we model fat as

being a slow and constant energy resource. Aerobic respiration in this model is dependent on

an excess of both oxygen and glucose, whereas anaerobic respiration does not depend on

oxygen and is in inhibited by the “Lactate Threshold” (Ghosh, 2004) as well as being

dependent on glucose. These qualitative characteristics are consistent with observation of the

limiting factors of running (Coe and Martin, 1991). An additional differential equation for oxygen

was modeled as such:

Oxygen was assumed to increase to a maximum amount during moderate to intense exercise

in a similar negative feedback paradigm, but also was decreased by aerobic respiration of both

fat and glucose. By modeling relationships with simple interactions informed by conservation

of energy and physiological relationships, we present a model with assumptions that are

reasonably justified and so should be considered critically.

Solution to Mathematical Models

This model was initially solved using Euler’s Method (see code section) by manipulating

variables to match the data given by Martin and Coe. Each component was able to be solved

separately to some extent, and then the solution was manipulated in a Bayesian inference to

match the data more accurately. First, the concentration of PC was set to an arbitrary

concentration and the rate constant of its use, r, was manipulated so that PC was used over

the first ten seconds. Then, the production term was added so that PC would regenerate at the

cessation of exercise as described by Clark (1997), but during exercise the concentration of PC

would be at a low equilibrium concentration. Then, the rate constant of anaerobic glucose

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(8)dO2

dt= a

O2max

�O2

O2max

+ 6G

aero

dt+ 6

dF

dta > 0

concentration, and the lactic acid effect on anaerobic respiration were manipulated so that

anaerobic respiration peaked at about one minute and then declined. Aerobic respiration was

then fitted in a similar way, first with oxygen assumed to be in excess and then with the oxygen

rate limiting term. Finally, the maximum oxygen term was set such that the overall energy

matched given data as closely as possible, and the rest of variables were manipulated in order

to match the data. By changing variables, observing the effect, and then keeping the change if

the result more closely matched the data or reverting to the previous state if the results

diverged from the data, we used Bayesian inference to match our model to the data.

An extension of this model attempted to take into account energy expenditures below

this world record level, where not all ATP would be consumed instantly, but instead would

accumulate and inhibit further ATP synthesis. This model used the chemical reactions and

assumed that the rate constants of these reactions caused the effects outlined in the previous

model. The best fits for this model had characteristics similar to those of the first model as

shown in Figure 2. This fitting was limited however by the fitting of the initial condition and the

size of the correction. Most random initial conditions and dynamics lead to unbounded growth

or decay, but even when the initial approximation is bounded and reasonable, Bayesian

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0 50 100 150 200 250 300time (seconds)

0

0.2

0.4

0.6

0.8

1C

ontri

butio

nPercent Contribution of Each Metabolite

PhosphocreatineAerobicAnaerobic

0 50 100 150 200 250 300time (seconds)

0

0.2

0.4

0.6

0.8

Ener

gy

Output of ATP and each component

ATPPCGaeroGanFat

Figure 1. ATP production was matched in the top left graph to the data from Martin and Coe (right). The bottom left graph shows the level of energy available over time predicted by this model and the contribution from each metabolite.

0 50 100 150 200 250 300 350 400 450Time (s)

0

10

20

30

40

50

60

70

80

90

% E

nerg

y C

ontri

butio

n

Governing Metabolic Pathways (100-3000m)

Phosphocreatine SystemAnaerobic RespirationAerobic Respiration

inference does not alway lead to a perfect fit. This is because most fits are local minimums for

error and not global minimums. Therefore, changing variables by a small amount cannot

decrease error because all local changes lead to an increase in error. Large changes are also

unlikely to lead to the best global fit because these are basically the same as randomizing the

initial conditions and variables; there is a global minimum but we do not know which variables

or starting conditions to manipulate to get there. This method is consequently limited in its

fitting ability, but has aspects that are qualitatively correct.

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Figure 2. Contributions of the metabolites to ATP in the Bayesian inference model.

0 0.5 1 1.5 2 2.5Time (Hundreds of Seconds)

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Cha

nge

in M

etab

olic

Con

cent

ratio

n

Bayesian Fit to the Three Metabolic Systems

GanGaeroATPPC

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Analytic Solutions to Mathematical Models

Phosphocreatine Pathway:

Under the assumptions that the ATP generation is predominantly governed by the phos-

phocreatine pathway during the first 30 seconds of exercise, we will model this pathway

independently of the others. Additionally, it is not influenced by the accumulation of lac-

tic acid, or the presence/absence of glucose or oxygen; it is solely reliant on the amount

of phosphocreatine available at the start of the exercise. As mentioned before, during a

resting phase ATP would be used to convert creatine back into phosphocreatine, e↵ectively

reversing the process. We are not particularly interested in modeling the metabolic path-

ways during rest, therefore we will assume the production term of phosphocreatine to be

zero. The di↵erential equation governing the amount of phosphocreatine in the system can

be simplified to:

dPC

dt= �rPC

where r is the reaction rate for the conversion of phosphocreatine into creatine, coupled

with the reaction of ADP into ATP. This single first-order di↵erential equation can be

solved directly by separation of variable:

dPC

PC= �rdt

Integrating both sides produces a solution of the form:

PC(t) = PC0e�rt

We can simplify this further by assuming that the concentration of phosphocreatine can

be normalized by the maximum concentration of PC:

PC(0) = PC0 = PCmax =) PC(t) = e�rt

1

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2

i.e. exponential decay.

Anaerobic Pathway:

As all of the phosphocreatine is converted into creatine, the phosphocreatine contributes

less and less to the overall ATP production. After about 30 seconds in the exercise state,

the anaerobic pathway becomes the dominant energy contributing pathway. The anaerobic

pathway is governed by the amount of glucose available. In order to simplify things further

we can assume that the muscle cells in question are connected to a continuous source of

glucose (from the bloodstream), therefore the limiting factor is the amount of lactic acid

present in the muscle. If the concentration of glucose in the cell is held constant, then

we can ignore the amount of glucose being burned in the anaerobic respiration pathway

and focus specifically on the amount of lactic acid being produced. We will denote the

anaerobic threshold as KLA, and the constant amount of glucose available [G] times the

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3

reaction rate h as g, therefore:

dLA

dt= g(1� LA

KLA) =

dATPan

dt

because every time a glucose molecule is converted into ATP, lactic acid is also produced.

In other words, they can be modeled by the same equation. This produced a first order

di↵erential equation of the form:

dLA

dt+ (

g

KLA)LA = g

Solving for the homogeneous solution first: (let k =

gKLA

)

dLAh

dt+ kLAh = 0

By separation of variable:

dLAh

LAh= �kdt

Produces a solution of the form:

LAh(t) = Ce�kt

For the particular solution, we can assume a solution of the form :

LAp(t) = a =) LA0p(t) = 0

Plugging this solution into the original di↵erential equation:

(0) +

g

KLA(a) = g =) a = KLA

Therefore:

LA(t) = LAh(t) + LAp(t) = C ⇤ e�kt+KLA,

if we assume LA(0) = 0, then:

LA(0) = 0 = C(1) +KLA =) C = �KLA =) LA(t) = �KLA ⇤ e�kt+KLA

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4

or

LA(t) = KLA ⇤ (1� e�kt)

When you plot the solution to this equation versus the %ATP production, during the rele-

vant time period, you notice that %ATP production begins to taper o↵ as LA approaches

KLA, the lactic acid carrying capacity. This is because further anaerobic respiration is

being inhibited by the presence of lactic acid. The drop in %ATP production that follows

is representative of the switch from anaerobic to aerobic respiration during this time, i.e.

the increase in %ATP production from aerobic respiration results in a decrease in %ATP

production from the anaerobic pathway.

Aerobic Pathway:

The simplifications made for the aerobic pathway are similar to those for the anaerobic

pathway. First we will assume that the concentrations of glucose and fat are similar to

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5

those in the nearby bloodstream. Furthermore, we will made the assumptions that these

quantities can be held constant, i.e. that the rest of the body can provide a constant supply

of glucose, G, and fat, F , to the muscle of interest. Combining these constant concentration

values with their respective reaction rates, i.e. k1 and k2, produces an equation for the

aerobic pathway is that fully dependent on the concentration ofO2 available. Let i represent

the rate at which O2 is delivered to the muscle, which is dependent on how much O2 is

present:

dO2

dt= i(1� O2

KO2

)� k1GO2 � k2FO2 = i� (

i

KO2

+ g + f)O2

Again, this will produce a non-homogeneous di↵erential equation of the form:

dO2

dt+ (

i

KO2

+ g + f)O2 = i

Solving for the homogeneous solution first:

dO2h

dt= �(

i

KO2

+ g + f)O2

Let c = (

iKO2

+ g + f), then by separation of variables:

dO2h

O2= �cdt

This produces a solution of the form:

O2h(t) = Ce�ct

Similarily to the anaerobic case, we assume a particular solution of the form:

O2p(t) = b =) O02p(t) = 0

Plugging back into the original di↵erential equation:

(0) + c(b) = i =) b =i

c

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6

Therefore:

O2(t) = O2h(t) +O2p(t) = Ce�ct+

i

c,

Assuming O2(0) = 0:

O2(0) = 0 = C(1) +

i

c=) C = � i

c

Finally:

O2(t) = � i

ce�ct

+

i

c=

i

c(1� e�ct

)

Discussion

The body’s metabolic system involves highly complicated interactions which can be

simplified to gain understanding about key attributes of this system, but by no means

understand it entirely. By building a model based on logical assumptions for the interaction of

metabolites and the reactants necessary for metabolism, we were able to qualitatively match

data about the body’s ability to produce ATP. The second graph in Figure 1 shows what we

would expect; that over longer periods of time with optimal expenditure of energy, the amount

of energy able to be expended decreases. As the saying goes, you can either go far or fast (in

some sense runners are going incredibly fast even in the marathon but relative to the 100m,

marathoners are going much slower). The contribution of PC is the greatest to start and then

decreases the fastest, the anaerobic metabolism starts lower and decreases slower and

aerobic metabolism starts the lowest and barely decreases at all, which is consistent with data

from Martin and Coe. This model also provides insight from fitting of initial conditions and rate

constants. The best fit model had initial conditions of PC, glucose and fat where glucose was

two orders of magnitude greater than PC to start, and fat was two orders of magnitude greater

than glucose. This is consistent with physiological observations of PC levels in muscles, blood

sugar levels and body fat content (Martins and Coe, 1991). Additionally, the rate constants

were best fit when PC use was three orders of magnitude greater than anaerobic metabolism,

which was three orders of magnitude greater than aerobic respiration. Finally, the maximum

oxygen production level was best fit when it was four times greater than the initial oxygen level,

which is somewhat consistent with observations of changes in breathing and heart rate (resting

heart rate is usually around 50 beats per minute (bpm) and peak performance heart rate is

usually around 200 bpm).

These results suggest applications for energy usage in both elite running and exercise,

and lifestyle and medicine. These data suggest that runners with stores of a particular

metabolite that are outside the normal proportional relationship may be especially well suited

to a particular event where they may have a physiological advantage. To maximize aerobic

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respiration relative to anaerobic respiration in order to increase efficiency and decrease lactic

acid, this model suggests that the oxygen concentration should be elevated to its maximum,

likely with the greatest effect before competition. In longer distance races, this model suggests,

in agreement with the data, that aerobic respiration and efficiency become the most important

factor to the limit that PC and anaerobic respiration become insignificant. This would suggest

even if people appear stronger, their success in long distance races may not be impacted

because they can only use a small percentage of their total available energy and consequently

only drive a small number of muscles over the course of the performance. In fact, muscle mass

may even be negatively correlated with success in distance running because it takes energy to

move muscles, which does not help performance if they are not being used. Finally, although

there are specific transitions in energy source, shown in the second part of Figure 1, the overall

ATP curve is monotonically decreasing. This would suggest that the optimal pacing strategy

would be to evenly expend energy throughout the performance (whereas if ATP expenditure

had a local maximum an optimal pacing strategy may include speeding up at that point). This

suggestion is confirmed by the pacing of world record races which are very evenly split (Martin

and Coe, 1991). This model suggests interesting insights that confirm more than refute

conventional wisdom. Further research of the metabolic system may lead to more revolutionary

implications.

Less conventional implications of this model may be applicable to medicine and

lifestyle, especially for treating stroke and obesity. Stroke occurs when blot clots prevent blood

from reaching the brain (ischemic stroke) or when blood vessels burst, causing blood to leak in

the brain and blood supply to be disrupted (hemorrhagic stroke). At the epicenter of a stroke

many neurons will die from the acute effects of the stroke, but much more of the damage is

done proximately where disrupted blood supply cause ion concentration deregulation and

consequently cell death. When blood supply is disrupted, ion pumps, which require energy to

function, fail to regulate ion concentrations (Ransom, 2016). This model suggests that

immediate supply of energy to these neurons could be provided by PC or anaerobic respiration

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for short periods of time if PC could be steadily supplied or lactic acid was able to be removed.

Emergency delivery and uptake system could add minutes to hours to the life of a neuron,

allowing doctors and the body time to reform necessary systems for aerobic respiration.

The same conditions necessary for aerobic respiration in the brain are necessary in the

body, which, in the later case, involves the burning of fat, a need for someone suffering from

obesity. Our model suggests that in order for people to burn more fat, they must have an

adequate supply of oxygen, and exercise for relatively long periods of time. This model

suggests that short periods of anaerobic energy use result in the depletion of PC and glucose

which would need to be resupplied after exercise. This would result in the body signaling for

energy to replenish these supplies, which would likely lead to other benefits of exercise but

probably less fat loss than if more of the energy use was aerobic. This model suggests that the

most effective technique for promoting the burning of fat for energy would be to sustain high

levels of oxygen.

By modeling the three systems of energy productions, we were able to qualitatively

match trends in the metabolism of elite runners and make inferences from this data that are

likely applicable. The complexity of these three systems makes this problem very difficult to

solve directly in an analytical way, and so we often rely on simplifications that allow us to study

key aspects of the problem. While it is possible that one day someone will create a complete

mathematical model of the body’s metabolic systems, more likely many people will make

useful models that will highlight pertinent relationships which will be more fruitful in

understanding how humans work. The ultimate goal of mechanistic models of complex

systems is often to determine the dynamics that underlie the mechanisms of the system and

provide evidence for the characteristics of the interactions of its parts. This has hopefully

enabled us to explain these systems in a more intuitive way.

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References

Clark, Joseph F (1997). “Creatine and Phosphocreatine: A Review of their Use in Exercise and Sport.” Journal of Athletic Training 32:1.

Fiske, C. H. and Y. Subbarow, “Phosphocreatine”, J. Biol. Chem., 81:629-679. (1929).

Freeman, Scott. “Biological Science.” 4E Benjamin Cummings (2010).

Goodwin, Matthew L., James E. Harris, Andrés Hernández, and L. Bruce Gladden (2007). “Blood Lactate Measurements and Analysis during Exercise: A Guide for Clinicians.” J Diabetes Sci Technol 1(4): 558–569.

Jenkins, Mark A., “Creatine Supplementation in Athletes: Review.” SportsMed Web. (1998). <http://riceinfo.rice.edu/~jenky/sports/creatine.html>

Martin, David E., Peter N. Coe. “Training Distance Runners.” Leisure Press (1991). Champagne, Illinois. 127.

Pretorius, Etheresia (2013). “The adaptability of red blood cells.” Cardiovascular Diabetology 12:63.

Powers, Scott, Edward Howley. “Exercise Physiology: Theory and Application to Fitness and Performance.” 9E McGraw Hill (2014).

Ransom, Christopher (2016). “Stroke: pathophysiology of cerebrovascular disease.” University of Washington. Lecture. Feb 26.

Solomons, T. W. Graham, Craig B. Fryhle, Scott A. Snyder (2013). “Organic Chemistry.” John Wiley & Sons Inc.

Thiele, Ines et. al. “A community-driven global reconstruction of human metabolism.” Nature Biotechnology 31:5. Nature America, Inc. (May 2013).

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Codehours = 0;

minutes = 5; minutes = minutes + 60*hours;

seconds = 0; seconds = seconds + 60*minutes;

dt = 1e-3; tfinal = seconds*1e6*dt;

dATP_dt = zeros(1,tfinal); dATP_dt(1) = 0;

PC = zeros(1,tfinal); PC(1) = 1e1; r = 1e2*dt*PC(1); p = 1e-5*dt*PC(1);

G = zeros(1,tfinal); G(1) = 1e3; c = 5e-4*dt*G(1); g = 1e-9*dt*G(1);

F = zeros(1,tfinal); F(1) = 1e5; d = 1e-4*dt*F(1);

LA = zeros(1,tfinal); LAmax = 8e0; b = 1e-1*dt*G(1);

O2 = zeros(1,tfinal); O2(1) = 1e4; a = 1e2*dt*O2(1); O2max = 4*O2(1);

dG_aero_dt = zeros(1,tfinal);

dG_an_dt = zeros(1,tfinal);

PCout = zeros(1,tfinal);

Fout = zeros(1,tfinal);

for t = 2:tfinal

PCout(t) = r *(PC(t-1)./(PC(1)+PC(t-1)));

PCin = p*(PC(1) - PC(t-1));

dPC_dt = PCin - PCout(t);

dG_aero_dt(t) = c*(O2(t-1)/(O2(1)+O2(t-1)))*(G(t-1)/(G(1)+G(t-1)));

Fout(t) = d*(O2(t-1)/(O2(1)+O2(t-1)));

dF_dt = -Fout(t);

dG_an_dt(t) = b*(G(t-1)/(1+G(t-1)))*((LAmax-LA(t-1))/LAmax);

dO2_dt = a *(O2max-O2(t-1))/O2max - 6*dG_aero_dt(t) - 6*Fout(t);

dLA_dt = 2*dG_an_dt(t);

PC(t) = PC(t-1) + dPC_dt*dt;

G(t) = G(t-1) - (dG_aero_dt(t) + dG_an_dt(t))*dt + g*(G(1)-G(t-1)); %glucose crash approximate

O2(t) = O2(t-1) + dO2_dt*dt;

F(t) = F(t-1) + dF_dt*dt*(F(t-1)/(F(1)+F(t-1)));

LA(t) = LA(t-1) + dLA_dt*dt;

dATP_dt(t) = PCout(t) + 32*dG_aero_dt(t) + Fout(t) + 2*dG_an_dt(t);

end

time = (1:tfinal)*dt;

figure

subplot(2,1,1);

plot(time,PCout./dATP_dt); hold on;

plot(time,(32*dG_aero_dt+Fout)./dATP_dt); hold on;

plot(time,2*dG_an_dt./dATP_dt);

title('Percent Contribution of Each Metabolite');

xlabel('time (seconds)'); ylabel('Contribution');

legend('Phosphocreatine','Aerobic','Anaerobic');

subplot(2,1,2);

plot(time,dATP_dt); hold on;

plot(time,PCout); hold on;

plot(time,32*dG_aero_dt); hold on;

plot(time,2*dG_an_dt); hold on;

plot(time,Fout);

title('Output of ATP and each component');

xlabel('time (seconds)'); ylabel('Energy');

legend(‘ATP','PC','G_{aero}','G_{an}','Fat');

�21

Alternate Model

close all

dt = 1e-2;

numtimes = 5; MAXNUMTIMES = 5;

% data %

wrtime = [10,20,43,102,209];

wrpc = [70,40,10,5,2];

wran = [22,46,60,38,22];

wraero = [8,12,30,57,76];

tfinal = max(wrtime(numtimes));

time = (1:tfinal)*dt;

E = 1;

numIC = 9; numvar = 8; numDEs = 11;

%load('x.mat'); %x = rand(1,numIC + numvar);

counter = 0; error = 1;

while(error > 0.75)

x = rand(1,numIC + numvar);

index = ceil(rand*length(x));

amount = (rand-0.5)/100;

if counter > 0

x(index) = x(index) + amount;

end

y = zeros(numIC,tfinal);

% PC,G,ATP,CO2,O2,H,Cr,LA,BR

y(:,1) = x(numvar+1:numvar+numIC);

dy_dt = zeros(numDEs,tfinal-1);

for t = 1:tfinal-1

dPC_dt = -x(1)*y(1,t)*y(6,t) + x(2)*y(3,t)*y(7,t);

dG_an_dt = -x(3)*y(2,t) + x(4)*(y(3,t)^2)*y(8,t)^2;

dG_aero_dt = -x(5)*y(2,t)*y(5,t)^6 + x(6)*(y(4,t)^6)*y(3,t)^32;

dG_dt = dG_aero_dt + dG_an_dt;

dATP_dt = -32*dG_aero_dt - dPC_dt -2*dG_an_dt - E*y(3,t);

dCO2_dt = -6*dG_aero_dt - x(8)*y(9,t);

dO2_dt = 6*dG_aero_dt + x(8)*y(9,t);

dH_dt = dPC_dt;

dCr_dt = -dPC_dt;

dLA_dt = -2*dG_an_dt;

dBR_dt = x(7)*dATP_dt;

dy_dt(:,t) = [dG_an_dt;dG_aero_dt;dATP_dt;dPC_dt;dG_dt; dCO2_dt; dO2_dt; dH_dt; dCr_dt; dLA_dt; dBR_dt];

y(:,t+1) = y(:,t) + dy_dt((numDEs-numIC)+1:numDEs,t)*dt;

end

e = 0;

for i = 1:numtimes

PC = dy_dt(4,wrtime(i)-1);

Gan = dy_dt(1,wrtime(i)-1);

Gaero = dy_dt(2,wrtime(i)-1);

ATP = dy_dt(3,wrtime(i)-1);

e = e + abs((wrpc(i)-(-PC/ATP))/wrpc(i)) + abs((wran(i)-(-Gan/ATP))/wran(i)) + … abs((wraero(i)-(-Gaero/ATP))/wraero(i));

end

errornew = e/3/numtimes;

if errornew < error

error = errornew

save x.mat x;

else

x(index) = x(index) - amount;

end

counter = counter +1;

if counter > 1e5

break

end

end

for i = 1:4

plot(time(1:end-1),dy_dt(i,:)); hold on;

end

legend('Gan','Gaero','ATP','PC');

�22

Analytic Code

data = load('Athelete data.txt');figure, hold onplot(data(1:6,2),data(1:6,5),'-o')plot(data(1:6,2),data(1:6,6),'-o')plot(data(1:6,2),data(1:6,7),'-o')xlabel('Time (s)')ylabel('% Energy Contribution')legend('Phosphocreatine System','Anaerobic Respiration','Aerobic Respiration','Location','Best')title('Governing Metabolic Pathways (100-3000m)')

%Phosphocreatine plottspan = 0:1:data(6,2);r=0.05;PC=exp(-r.*tspan);

figure, hold onplot(data(1:6,2),data(1:6,5)/100,'-o')plot(tspan,PC)title('Phosphocreatine Qualitative Behavior, r=0.05')xlabel('Time (s)')ylabel('%Energy Contribution’)

%Anaerobictspan = 0:1:data(4,2);k=0.05;LA_max=.7;LA=LA_max*(1-exp(-k.*tspan));figure, hold onplot(data(1:4,2),data(1:4,6)/100,'-o')plot(tspan,LA)title('Anaerobic Qualitative Behavior, k=0.05')xlabel('Time (s)')ylabel('LA Production')legend('%ATP Production from Anaerobic','LA Accumulation’,'Location','Best')

%Aerobictspan = 0:1:data(9,2);i=0.1;c=0.005;O2=(i/c)*(1-exp(-c.*tspan));figure, hold onplot(data(1:9,2),data(1:9,7)/100,'-o')plot(tspan,O2./max(O2))title('Aerobic Qualitative Behavior, i=0.01, c=0.005')xlabel('Time (s)')legend('%ATP Production from Aerobic','ATP from O_2(t)','Location','Best')

�23

Athlete data.txt100 10 11 Nan 70 22 8

200 20 22 Nan 40 46 14

400 43 48 Nan 10 60 30

800 102 113 135 5 38 57

1500 209 232 112 2 22 76

3000 449 503 102 0 12 88

5000 778 877 97 0 7 93

10000 1628 1814 92 0 3 97

42195 7610 8466 82 0 0 99