Download Modelling glycolysis with Cellware

Mikko Laiterä
63707A
Helsinki University of Technology
Basics for the Biosystems of the Cell
S-114.2500
Exercise
Modelling glycolysis with Cellware
Contents
1.
Abstract
2.
2.1.
2.2.
2.3.
Glycolysis
Introduction to glycolysis
Reactions and stoichiometry of glycolysis
Regulation of glycolysis
3.
3.1.
3.2.
3.3.
Cellware
Introduction to Cellware
Reaction kinetics applied in the model
Mathematical methods for simulation
4.
4.1.
4.2.
Modelling glyoclysis with Cellware
Constructing a model
Simulation of glycolysis
5.
Discussion
6.
References
2
1.
Abstract
Recent advances in biological sciences and information technology have made the development of
“computer assisted biology” possible and the research performed this way is often referred to be
done “in silico”. Computers were initially used mostly for storing information but their capability to
solve numerical problems soon opened new opportunities to apply their rapidly increasing power to
understand the phenomena in organisms. This development also urged the need to formulate
physical and chemical laws of metabolism in the language of mathematics. The particular approach
in which models are constructed to cover whole biological networks is known as systems biology.
The term “glycolysis” is derived form Greek words glyk meaning sweet and lysis for dissolution. In
consistent with this logic, glycolysis is the sequence of reactions that metabolizes one molecule of
glucose to two molecules of pyruvate with the concomitant net production of two molecules of ATP
[1]. Glycolysis is employed by a great variety of organisms, both aerobic and anaerobic, making it
the most common metabolic reaction pathway in living things. In aerobic species the two pyruvate
moleucles are further oxidised into carbon dioxide (CO2) in citric acid cycle and their reducing
power harnessed in form of activated carriers of electrons (FADH2 and NADH) to faciliate
oxidative phosphorylation. In anaerobic conditions pyruvate is converted to ethanol or to lactate
(e.g. in muscle cells during anaerobic exercise). The degree of glycolysis or gluconeogenesis,
synthesis of glucose from pyruvate, employed in a cell is also tightly controlled. The goal of chapter
two was to review glycolysis profoundly enough to introduce basics needed to understand the
model constructed and results of the simulation represented in chapter four.
The Cellware program is a non-commercial graphical systems biology tool for modelling and
simulation. It is rather easy to use after learning to avoid few irritating bugs. The program is also
designed to serve scientists with little programming experience. Probably the user-friendliness of
the program is its most important feature. To solve differential equations derived from laws of
physical chemistry the Cellware simulation engine is harnessed with classical algorithms (e.g. 4th
order Runge-Kutta) and a few interesting modern methods. Also for phenomena including
randomness stochastic algorithms are available. The chapter three concerning Cellware is mostly
about theoretical features not explained in the Cellware manual. These subjects are also applicable
in other systems biology tools in general.
The chapter four is about modelling glycolysis with Cellware. The objective was to show that in
silico methods can faciliate better understanding of biological phenomena and even produce new
information. The most difficult problem in modelling glycolysis with the Cellware was to acquire
suitable equations and parameters to describe each reaction. Fortunately, the efforts were rewarded
and the results of the simulation gave insight to the qualitative variation of concentrations of
metabolites during reaction. Probably more importantly the simulation supported the common
practice of dividing glycolysis into two different phases. Also at least one reference to the allosteric
control in the model of glycolysis was discovered.
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2.
Glycolysis
2.1. Introduction to glycolysis
Glycolysis is also known as the Embden-Meyerhof pathway named after two German scientists for
their research on subject. The essence of this reaction pathway is to create energy for cell’s needs:
when one molecule of glucose is converted to two molecules of pyruvate energy is stored in
phosphorylation of two molecules of ADP to two molecules of ATP and in reduction of two
electron carriers (2 NAD+ + 2e- → 2 NADH). It is important to stress on the fact that glycolysis
itself is an anaerobical pathway. In consequence of this one might guess, that glycolysis is ancient
in evolutical way. Actually glycolysis was the first metabolic pathway to develop in evolution [3].
In eukaryotes glycolysis takes place in the cytosol. Another point to note is that glycolysis follows
quite a forward pattern, unlike e.g. citric acid cycle, which is affected not only by energy charge
(energy charge = ([ATP]+(1/2) × [ADP]) / ([ATP]+[ADP]+[AMP])) and pyruvate concentration but
also e.g. by fatty acid synthesis (from citrate) and amino acid syntesis (from oxaloacetate).
Glycolysis is nearly a straight path. To mention some important exceptions fructose and galactose
may enter the reaction pathway: fructose can be phosphorylated to a glycolysis metabolite fructose
6-phosphate by hexokinase, although hexokinase has about 20 times greater affinity for glucose.
Galactose can be converted to glucose in four reaciton steps. Also a glycolysis metabolite
phosphoenolpyruvate can lead to amino acid synthesis.
Picture one: Simple representation
of glycolysis [4]
2.2. Reactions and stoichiometry of glycolysis
Glycolysis consists of ten reactions and eleven metabolites. Each reaction is catalysed by a different
enzyme. The glycolysis can be divided into two or three different phases. A picture of the pathway
divided into two phases is shown below. The first phase is called a Preparatory phase, although one
might prefer the term “Investing phase” to emphasize the fact that in this first phase energy is spent
to gain “profit” later. The second phase is defined here as the “Payoff phase”.
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Picture two [5]: Reactions of glycolysis. It should be stressed that when starting with one
molecule of glucose there are two molecules of a metabolite in each step of the Payoff phase.
5
Overal net reaction of glycolysis is
glucose + 2 Pi + 2 ADP + 2 NAD + → 2 pyruvate + 2 ATP + 2 NADH + 2 H + + 2 H 2 O (2.2.1),
where Pi is orthophosphate HPO32-, in which the negative charge is delocalised to oxygen atoms.
NAD+ is Nicotinamide adenine dinucleotide and can be reduced to NADH by accepting two
electrons and a proton. Reaction to gain NADH can be called a ‘hydride transfer’, but it must be
kept in mind that electrons of a breaking bond (Highest Occupied Molecular Orbital, HOMO)
attack NAD+ (which has Lowest Unoccupied Molecular Orbital, LUMO) and no pure H- occurs in
reaction.
The most important reactions should be considered a little more in detail. In the first reaction
enzyme hexokinase phosphorylates glucose and one ATP is consumed (to ADP). This reaction is
notable for three reasons: 1) Hexokinase is one of the regulated enzymes in the pathway, 2)
phosphorylation traps glucose in the cell because of two negative charges in glucose 6-phosphate
and 3) Phosphorylation destabilizes glucose making way for further catalysis. Third reaction in
glycolysis is also a phosphorylation reaction. In this reaction the enzyme phosphofructokinase
(PFK) catalyses the phosphorylation of fructose 6-phosphate to fructose 1,6-biphosphate. PFK is
the key regulation enzyme in glycolysis and sets the pace of the reaction pathway. An interesting
reaction is also fifth reaction of the pathway. It is an isomerization reaction between
Dihydroxyacetone phosphate (DAP) and Glyseraldehyde 3-phosphate (GAP) catalysed by enzyme
Triose phosphate isomerase (TIM), which is a kinetically perfect enzyme. This means that the speed
of catalysis is restricted by diffusion rate rather than by the capasity of the enzyme: TIM accelerates
this reaction about 1010-fold. At equilibrium, 96% of the triose phosphate is dihydroxyacetone
phosphate [1], but because only GAP can react further in the glycolytic pathway and is thus
consumed, the equilibrium is never reached and no DAP is left over.
The sixth reaction belongs to the payoff phase. By enzyme Glyseraldehyde 3-phosphate
dehydrogenase GAP is converted to 1,3-biphosphoglycerate (1,3-BPG). Actually this reaction
consists of four steps (in literature sometimes three, because reduction of NAD+ is not always
counted) and has a thioester intermediate. However, the important issue to understand is that this
reaction produces a molecule (1,3-BPG) with high phosphoryltransfer potential i.e. a molecule that
can phosphorylate ADP to ATP is formed. It should also be noticed that the reaction consists of an
oxidising step, where the reaction intermediate is oxidised and NAD+ is reduced to NADH. To
maintain the redox equilibrium in a cell the NADH formed must be oxidised later. As maybe easily
guessed, in the following reaction (7th) 1,3-BGP phosphorylates one ADP to ATP. The last reaction
to mention (10th) is also the last reaction of glycolysis: An another molecule of high
phosphorylation potential, phosphoenolpyruvate (PEP), is converted to pyruvate and ATP is formed
from ADP. Pyruvate kinase catalyses this reacion and is last (third) enzyme in the reaction pathway
subjected to allosteric control.
To represent the stoichiometry of glycolysis, the values of thermodynamic equilibrium constants are
needed. One should notice the connection between the manner of a reaction and the name of the
enzyme catalysing this reaction. If the first reaction in the table has ∆G = -33,5 kJ/mol [1], the
equilibrium constant would be K eq = e − ∆G / RT ≈ 403000 , where the molar gas constant
R ≈ 8,3145 J / Kmol and physiological temperature T = 312.15 K. However in physiological
conditions this value is K eq ≈ 250 [8], i.e. the reaction is not considered irreversible. For an
example equilibrium constant values in skeletal muscle tissue are represented are below [1], [3]:
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Table one: Stoichiometry of glycolysis in skeletal muscle tissue [1], [3], [8]
∆G (kJ/mol) in
Reaction
Enzyme
1. Glucose + ATP → glucose 6-phosphate + ADP +
H+
2. Glucose 6-phosphate → fructose 6-phosphate
3. Fructose 6-phosphate + ATP →
fructose 1,6-biphosphate + ADP + H+
4. fructose 1,6-biphosphate → dihydroxyacetone
phosphate + glyseraldehyde 3-phosphate
5. Dihydroxyacetone phosphate → glyseraldehyde 3phosphate
6. Glyseraldehyde 3-phosphate + Pi + NAD+ → 1,3biphosphoglycerate + NADH + H+
7. 1,3-Biphosphoglycerate + ADP → 3phosphoglycerate + ATP
8. 3-Phosphoglycerate → 2-phosphoglycerate
9. 2-Phosphoglycerate → phosphoenolpyruvate +
H2O
10. Phosphoenolpyruvate + ADP + H+ → pyruvate +
ATP
Hexokinase
Phosphoglucose
isomerase
Phosphofructokinas
e
Aldolase
Triose phosphate
isomerase
Glyceraldehyde 3phosphate
dehydrogenase
Phosphoglycerate
kinase
Phosphoglycerate
mutase
Enolase
Pyruvate kinase
physiological
conditions
-33.5
Thermodynamic
equilibrium
constant
250
-2.5
0.45
-22.2
242
-1.3
9.5 × 10
+2.5
0.052
-1.7
0.089
+1.3
57109
+0.8
0.18
-3.3
0.49
-16.4
10304
−5
The product of equilibrium constants is bigger than one, which means that the reaction will proceed
“right”. Quantitative analysis of glycolysis will be considered more profoundly in chapter four.
2.3. Regulation of glycolysis
The key point in understanding the glycolytic pathway is to recognize the control sites of the
pathway i.e. to remember which enzymes are subjected to allosteric regulation and to understand
how this is achieved. The activity of glycolysis (or more precisely; its enzymes) is adjusted to meet
the energy needs of the cell, but also the need of “building blocks” e.g. for amino acid syntesis. In
the reverse pathway, gluconeogenesis, most of the enzymes are same: only the regulatory steps
differ. As gluconeogenesis consumes six ATP equivalents (4 ATP, 2 GTP) to convert two
molecules of pyruvate to one molecule of glucose, it is essential that the two opposing metabolic
pathways do not occur simultaneously (possible exception is brown fat).
Powerful regulation also poses a problem for modelling: As some enzymes are allosterically
controlled they do not display the Michaelis-Menten kinetics. This means that to construct a
sensible model to give meaningful output, lots of research and calculation is needed. Another
problem is the import of new glucose into the cell and the consumption of pyruvate. In a model an
equilibrium would be reached unlike in real cells.
Glycolysis is controlled at three sites. Table one shows that biggest drops in Gibbs free-energy are
catalysed by hexokinase, phosphofructokinase and pyruvate kinase respectively. Not suprisingly,
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precisely these enzymes are regulated by reversible binding of allosteric effectors or by covalent
modification.
Depending on the organism, the process to gain energy for the cell often starts earlier than in
glycolysis. For example, hormonal response might lead to breakdown of glycogen, which is
followed by glycolysis. However, the phosphorylation of glucose by hexokinase is the first reaction
in the glycolytic pathway. This reaction can also lead to formation of glycogen or to pentose
phosphate pathway. After this first step considerable amount of energy is invested and glucose 6phosphate is trapped inside the cell. Perhaps suprisingly simply, hexokinase displays feedback
inhibition: evolving glucose 6-phosphate inhibites further phosphorylation. Also the substrate for
PFK, fructose 6-phosphate, is in equilibrium with glucose 6-phosphate, which means that inhibition
of PFK leads to evolving of glucose 6-phosphate and thus to inhibition of hexokinase.
Phosphofructokinase is the most important control element in the mammalian glycolytic pathway
[1]. Although hexokinase catalyses the first step of glycolysis PFK catalyses the committed step,
which is the first reaction unique to glycolysis. Similar to hexokinase the product of the reaction,
fructose 1,6-biphosphate, inhibites PFK. Phosphofructokinase is also very sensitive to energy
changes in the cell: ATP is an allosteric inhibitor and AMP an allosteric activator of this enzyme.
Glycolysis is sometimes followed by the lactic acid fermentation. To prevent the excessive
formation of lactatic acid the low pH, i.e. high H+-consentration, also inhibites PFK. In aerobic
organisms the pyruvate is at normal state usually (e.g. not in erythrocytes) transported to citric acid
cycle. Thus it is rational that citrate, which is a metabolite of the Krebs cycle is also an inhibitor to
PFK.
Pyruvate kinase catalyses the final reaction of glycolysis. Abundance of fructose 1,6-biphosphate,
which is the product of the reaction catalysed by PFK, activates pyruvate kinase. This can be seen
as a signal to carry on the payoff phase when “the investment” is already made. Similarily to this
logic, ATP is a signal of high energy charge in the cell and presence of alanine, which can be
synthesized in one step from pyruvate, suggests that the cell has sufficient amount of “building
blocks”. Both substances mentioned are allosteric inhibitors of pyruvate kinase.
3.
Cellware
3.1. Introduction to Cellware
Built with Java and C++ and developed by Systems Biology Group of Singapore Bioinformatics
Institute, the Cellware program (30.5.2006 current version is 3.0.1) is free for academic use. The
program itself is designed to be a modelling and simulation platform for systems biology, which
means that the Cellware is not only capable of processing complicated reactions of biochemistry but
also large networks of reactions. The Cellware program also includes analysis methods e.g.
parameter estimation. Variety of both stochastic and deterministic algorithms for simulation
faciliate a range of possibilities to construct models.
Importantly, the Cellware is rather user-friendly: it is compatible with Windows (unlike many other
biological programs) and it has a graphical user interface. This includes program’s ability to
construct graphical representations of substance consentrations versus time during reactions. The
Cellware also supports import and export of models in the System Biology Mark-up Language
(SBML) file format [2]. Comprehensive user manual with examples included help to understand the
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program. As this chapter is not ment to teach one to use Cellware, the program manual should be
consulted if the reader is interested in using the program.
When a model for phenomenon is needed the understanding of elementary reaction kinetics is
helpful. One must also choose a suitable algorithm for each simulation and thus the mathematics
applied by the Cellware engine should be examined. The subjects to discuss in the next section deal
with the phenomena encountered when modelling glycolysis. Thus the review of kinetics and
mathematical biochemisty does not cover great many essential features.
Picture three: Cellware “New Project”-window
3.2. Reaction kinetics applied in the model
Kinetics shall be considered in a rather basic level with a pedagogical point of view. A chemical
reaction can be expressed:
aA + bB + ... → cC + dD + ... (3.2.1),
where uppercase letters indicate a chemical species and each lowercase letter represents the
stoichiometric coefficient of a species. Let us consider an example:
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4 NO2 ( g ) + O2 ( g ) → 2 N 2 O5 ( g ) .
First and foremost the mass action law states that if a system is at equilibrium at a given
temperature, the thermodynamic equilibrium constant
K=
[ C ]c [ D ]d ...
[ A ]a [ B ]b ...
(3.2.2),
where [i] is a concentration of a species i at equilibrium. The example would give K =
[ N 2O5 ]2
[ NO2 ]4 [ O2 ]
. An
important relationship, which can derived using laws and definitions of thermodynamics, links the
equilibrium constant and the change of Gibbs free energy in reaction:
∆G = − RT ln K (3.2.3),
where “lnK” is the natural logarithm of the equilibrium constant.
Generally during a reaction the number of a species i can be denoted
ni = ni0 + vi ξ (3.2.4),
where ni is the number of moles of species i at any given time durig the reaction, ni0 is the number
of moles of a species in the beginning of the reaction, vi is a stoichiometric coefficient of a species i
(negative if i is a reactant, positive if i is a product) and ξ is the advancement of the reaction (equal
to zero in the beginning).
By differentiating both sides from equation (3.2.4) with respect to time is acquired
dni / dt = vi dξ / dt (3.2.5).
The reaction rate is defined as Rate = dξ / dt . Equation (3.2.5) ⇒
Rate =
1
vi
dni / dt (3.2.6).
In the example Rate = − 14 dn NO2 / dt but Rate = 12 dn N 2O5 / dt would do as well.
The problem with equation (3.2.6) is that it depends on the size of the system, i.e. it is an extensive
property. The consentration of a species i in a system is [i ] = n / V ⇒ d [i ] = d (n / V ) and thus a more
convenient intensive quantity can be used:
R = Rate / V =
1 d
vi dt
( ni / V ) =
1
vi
d [i ] / dt (3.2.7).
To define R it was assumed that the volume of the system (V) is independent of time, which is quite
logical especially if the time interval is short.
So called the rate law is also important. The intensive reaction rate can be written as
10
R = k ∏ [ Ai ]α i (3.2.8),
i
where k is referred to as the rate constant of the reaction and α i is the reaction order with respect to
species Ai. The sum Σ α i is called the overall reaction order. This may seem as an easy way to
i
predict reaction rates but the difficulty is that the parameters αi and k must be determined
experimentally. With the example mentioned previously R = k[ NO2 ] 2 [O2 ] is valid.
The next result shall be used in chapter four. If a reaction reaches an equilibrium, it must have a
constant reaction rate also to the reaction from products to reactants. Let the forward rate reaction
constant be ka and for opposite direction kb. To simplify matters the forward reaction is first order in
A and the back reaction is first order in B: A ↔ B (apologies for wrong type of arrow between A
and B). Because of the dynamic equilibrium, d [ A] / dt = − k A [ A] + k B [ B] = d [ B ] / dt = k A [ A] − k b [ B] .
This can be rearranged to 2k A [ A] = 2k B [ B ] and finally (kB, [A] ≠ 0) to
k A / k B = [ B] /[ A] = K (3.2.9)
by definition of the equilibrium constant. Thus the equilibrium constant K is linked to reaction
kinetics.
Quantitative biochemisty consists of a wide range of experimentally derived equations. One of the
most important mathematical tools in enzyme kinetics is the Michaelis-Menten equation:
v0 = v max
[S ]
[ S ]+ K M
(3.2.10),
where v0 is the initial rate of the reaction, vmax is the maximum rate of the reaction, [S] is the
substrate consentration and KM is experimentally determined Michaelis constant. To derive this
equation so called steady-state conditions must be assumed. This means that even though the
concentrations of intermediates of the reaction vary, the concentration of the enzyme-substrate
complex is constant [ES]. An other way to put this is that the rates of formation and breakdown of
the [ES] complex are equal. It is also assumed that almost no product reverts to the initial substrate.
To be precise the allosteric enzymes do not obey Michaelis-Menten equation. However, so-called
pseudokinetic rate constants Vmax and Km can be calculated for allosterically controlled enzymes.
These constants offer approximate quantification of enzyme kinetics in a narrow predefined
substrate consentration interval.
3.3. Mathematical methods for simulation
When bioprocesses are simulated, different reactions obeying different kinetics conjugate to form a
network of reactions or at least a reaction pathway. One reaction may have an impact on several
other reactions and vice versa. These reactions must be written in the language of mathematics if
quantitative analysis is needed. Kinetics and thermodynamics provide the means for this. As some
variables are unknown and the laws of physics tend to be in a form of differential equations lots of
data must be handled in a short time. In addition, most differential equations cannot be solved
11
explicitly and thus numerical methods are needed. This often requires lots of work, i.e. the program
must solve many calculations. To reduce this calculation work more efficient algorithms are
needed. Developing algorithms, which reduce the number of iterations required or simplify the
problem, is a science of its own right. The simulation engine includes two different classes of
algorithms, namely the Stochastic Algorithms and Deterministic Algorithms [2]. In addition one
hybrid algorithm consisting features from both classes can be chosen.
Deterministic algorithms are used when the problem lacks randomnes or it can be ignored. These
algorithms are constructed to solve problems that can be formulated using laws of kinetics.
These algorithms are convenient for solving problems related to enzyme kinetics and thus they are
suitable for simulations of metabolism. The accuracy of the result depends on three things: A
function can be “tame”, which means in common language that it doesn’t make rapid turns or have
sharp peaks. Some functions are more difficult, and the values of the function and its derivatives
may change a lot at some time intervals. Secondly, some numerical methods are simply more
effective than others. Finally, the step size is practically the most important factor. The step size is
the interval (usually time) between two evaluation points. Thus, the smaller the step size, the
smaller the error. Normally one must optimize the method and step size to maximize the accuracy,
while maintaining a reasonable simulation time (proportional to calculation work). As most of the
differential equations to encounter cannot be solved explicitely the three methods offered in
Cellware are all numerical. Actually, giving a “symbolic” result is very difficult for programs as
they normally consider numbers as “floats” (floating desimal number). The three possible methods
are:
1) Euler forward ode (ode means ordinary differential equation). Euler forward can be
considered the most simple algorithm to solve differential equations and is derived below as
an example. However, no proof of convergence towards the right solution will be shown.
The problem is solved with respect to one variabe. An ODE to solve with respect to time can
be formulated
x' (t ) = f (t , x(t )) , t ∈ [0, T ] , x(0) = x 0 (3.3.1).
The last statement is the initial condition and x0 is a real number. First let q be a continuous
function to be solved with respect to time t. If the function q depends also on a set of other
variables (e.g. substrate constentration, pH), they can be all represented as one vector y (or if
they are equations too a matrix can be used). Now, y has a dimension equivivalent to
number of these “other” variables. A continuous function can be represented as its Taylor
series:
∞
q ( y, t ) = ∑ q
k =0
(k )
( y ,t 0 )
k!
(t − t 0 ) k (3.3.2),
where t 0 is a predefined point for this development (often zero) and q ( k ) ( y, t 0 ) means the kth
derivative of q with respect to t in t 0 . Taylor series always converge (readers interested in
proofs should consult literature of analysis), which can be guessed due to rapidly growing
divisor k!. Because of this convergense first terms in the serie have greater effect on the sum
than the following terms. Thus x of the original ODE (3.3.1) can be approximated by first
two terms:
x(t ) ≈ x(t 0 ) + [ dtd x(t 0 )](t − t 0 ) (3.3.3).
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In numerical methods the time is discreted into steps t0, t0+h, t0+2h,... until the predefined
end (e.g. solve consentration after 10s has passed) is reached. The step size is h. To simplify
and shorten notations (not the matter!) is assumed t0 =0, i.e time measuring starts at the
same time with the simulation. Also in first order differential equations dtd x(t ) is known as a
function of x and t. Thus it can be referred to as f (t , x(t )) . In the explisite Euler method the
iterate from previous round is a seed for the next round: x 0 = x(0) , which is a known
quantity and called an initial condition. The first iterate is
x1 = x 0 + f (0, x(0))h = x 0 + f (0, x 0 ) and the second x 2 = x 1 + f (h, x1 )h . Thus the
algorithm can be written (k is the number of the iterate).
x k +1 = x k + hf (hk , x k ) k = 0,1,2,.. (3.3.4).
2) Runge-Kutta 4th order is a classic for an algorithm to solve differential equations. The order
means phases vi calculated per one step. Euler forward is of first order, which suggests that
Runge-Kutta 4th must be rather more complicated. However, Runge-Kutta 4th has a local
error of factor four (the local error is proportional to fourth power of the step), while Euler
forward has a local error of factor one. Runge-Kutta 4th has also better accuracy for
calculation work done. Thus Euler forward must be preferred only if the amount of
calculation needed is too numerous for Runge-Kutta 4th, although even when dealing with
such a massive amount of data should Runge-Kutta 4th with longer step (than for Euler) be
considered. No properties for this method is proved here and interested readers should again
refer to the literature of analysis. With a common shortening from literature hk = tk and with
the similar notations to 1) the algorithm for Runge-Kutta 4th is:
v 1 = f (t k , x k ),
i −1
v i = f (t k + ci h, x k + h∑ aij v j ), i = 2,...,4
(3.3.5)
j =1
4
x k +1 = x k + h∑ bi v i
i =1
Thus the phases vi are calculated recursively and the new iterate xk+1 is calculated using
phases and the iterate xk from the previous round. The coefficients for Runge-Kutta 4th are
c1 = aij = 0, except a21 = a32 = ½ , a43 = 1, b1 = b4 = 1/6 , b2 = b3 = 2/6, c2 = c3 = ½ and
c4 = 1.
3) Cellware has also an Advanced ODE Solver for “stiff” ODEs. ODE can be called stiff if it
has both tame and difficult sections at chosen time interval. Functions with rapidly changing
derivatives require lots of evaluations (calculations) to be made but if a short time step is
applied the simulation might take a very long time. Thus stiff ODEs would require short
time step, but calculation would take time because of long tame sections of the function.
Actually Advanced ODE Solver does not refer to any particular algorithm. Possibly
literature has been consulted and an effective algorithm for solving stiff ODEs was chosen.
However, this algorithm may have been developed by the Systems Biology Group of
Bioinformatics Institute of Singapore itself and it may even be patented. Usually these
algorithms for stiff ODEs evaluate derivatives of the function to solve and use shorter time
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steps when function has rapidly changing derivatives and save calculation time by using
longer steps when the function acts “nicely”. Many of such algorithms are somehow based
on Runge-Kutta methods.
The Cellware simulation engine includes also stochastic algorithms. Shortly, these algorithms
should be used (only), when the simulation consists of random factors. For example, if radioactive
decays of a very small quantity of substace are calculated to happen nine times in a minute on
average this does not mean (deterministically) that in every minute nine decays take place. Most
common situation to require stocastic algorithms is a model dealing with very small consentrations
of substances. In these situations the random kinetic energy and movement of single molecules
must be considered. Also as a small number of molecules cannot be considered to be evenly
distributed, the consentration ceases to be a meaningful quantity to measure. However, glycolysis is
a well defined metabolic pathway with sufficiently large consentrations and thus deterministic
algorithms are used in the simulation. For this reason stochastic algorithms are not considered
further here. A reader interested in stochastic methods of Cellware should refer to its User Manual.
4.
Modelling glyoclysis with Cellware
4.1. Constructing a model
In the Cellware program “New Project” must be chosen. Then all eleven metabolites can be put on
their places. To clarify the model circles representing metabolites are named. Note that the
“Caption” of a metabolite cannot contain spaces. Although most properties function fine in version
3.0.1, it is not possible to give a species unique “Full name”. Rather, the program gives always the
same name for each species. However, this is quite small a problem as only the “Caption”, which
works fine, is shown. Drawing reaction arrows between metabolites (including ATP, NAD, etc.) is
a bit messy. Thus the biomolecules, which only accept or lose electrons or an orthophosphate are
coloured red as well as the orthophosphate itself. It is essential to remember that in an organism the
glycolytic pathway is always followed by reaction(s) to maintain the redox balance, i.e. NADH
produced must be oxidised back to NAD+. Thus, in model the amounts of NADH and NAD+ are
fixed.
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Picture four: The arrows indicate whether a metabolite is consumed or produced in reaction.
Unfortunately font of the text cannot be changed and thus the names of the metabolites show poorly
in picture.
4.2. Simulation of glycolysis
Simulation of glycolysis requires initial concentrations of metabolites and kinetic parameter values
for reactions. The glycolytic pathway is to be modelled in a yeast cell and suitable kinetics are
chosen for each enzyme separately. The program sometimes runs simulations totally wrong if a
saved model is opened. Thus the simulation is preferred to be run right after reactions are defined.
For reactions including only one substrate and one product the realationship
k A / k B = [ B] /[ A] = K (3.2.9)
derived in chapter 3.2. is used. This is done by setting the reverse rate constant one. It should be
stressed that this formula is quite approximative as it assumes first order kinetics. For reactions
involvig more substrates or products Michaelis-Menten kinetics (3.2.10) or Mass-action kinetics
(3.2.8) is used. A slight problem occurred when Cellware refused to apply Michaelis-Menten
kinetics for reactions involving more than one substrate or product. The problem was circulated by
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constructing the Michaelis-Menten equation manually. Below are concentrations of glycolysis key
metabolites in a yeast cell represented in a table:
Table two [10], [11]
Metabolite
Glucose
Glucose 6-phosphate
Fructose 6-phosphate
Fructose 1,6-biphosphate
Dihydroxyacetone phosphate
Glyceraldehyde 3-phosphate
1,3-Biphosphoglycerate
3-phosphoglycerate
2-phosphoglycerate
phosphoenolpyruvate
pyruvate
ATP
ADP
NAD+
NADH
Pi
Concentration (mM)
100
2.45
0.62
5.51
0.81
0.15
0.000583
0.52
0.07
0.08
1.85
2.52
1.29
1.55
0.004
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Caption used
GLUC
G6P
F6P
FBP
DHAP
GAP
BPG
PG3
PG2
PEP
PYR
ATP
ADP
NAD
NADH
P
In the second table of this section are mentioned kinetics used for each reaction and values of
kinetic parameters required [10], [12], [13]:
Table three: The kinetic parameters and pseudokinetic rate constants adjusted for conditions in a
yeast cell.
Kinetics
Michaelis-Menten
Mass action
Reaction
(mM, µmol/min)
MM
1. Glucose + ATP → glucose 6-phosphate + ADP +
H+
MA first order
2. Glucose 6-phosphate → fructose 6-phosphate
3. Fructose 6-phosphate + ATP →
fructose 1,6-biphosphate + ADP + H+
4. fructose 1,6-biphosphate → dihydroxyacetone
phosphate + glyseraldehyde 3-phosphate
5. Dihydroxyacetone phosphate → glyseraldehyde 3phosphate
6. Glyseraldehyde 3-phosphate + Pi + NAD+ → 1,3biphosphoglycerate + NADH + H+
MA first order
MM
MM
(mM)
MA
k(GLUC) = 0.08
k(ATP) = 0.15
k(G6P) = 30
k(ADP) = 0.23
Keq = 2.637
Km = 0.0471
Vm = 0.68
Km = 0.045
Vm = 1.19
MA first order
Keq = 0.379
MA first order
k(GAP) = 0.21
k(NAD) = 0.09
k(BPG) = 0.0098
k(NADH) = 0.06
k(BGP) = 0.003
k(ADP) = 0.2
k(PG3) = 0.53
k(ATP) = 0.3
Keq = 0.733
k(PG2) = 0.04
k(PEP) = 0.5
7. 1,3-Biphosphoglycerate + ADP → 3phosphoglycerate + ATP
MA first order
8. 3-Phosphoglycerate → 2-phosphoglycerate
9. 2-Phosphoglycerate → phosphoenolpyruvate +
H2O
MA first order
MA first order
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10. Phosphoenolpyruvate + ADP + H+ → pyruvate +
ATP
MA first order
k(PEP) = 0.14
k(ADP) = 0.53
k(PYR) = 21
k(ATP) = 1.5
The resulting graph from simulation looked a bit mysterious but after a little consideration it began
to make sense.
Picture five: Consentrations of some key metabolites of the glycolytic pathway using 5.0 sec
simulation time.
In a picture above, lots of oscillation is shown. First important observation is that the model clearly
is not sufficient to cover very long simulation times. This is due to consentrations and kinetic
parameters being constants. The situation where no ATP is consumed and no excess glucose is
imported is unnatural. According to the picture, the model is valid for about 1.8 seconds, which is
actually quite long time for a static model of glycolysis. Also the Cellware simulation engine does
not function perfectly as no consentration should ever be less than zero. Maybe this phenomenon
should be interpretated as a situation where virtually all metabolite intitially present is consumed
and also some imported quantity of this metabolite (if a metabolite is ‘borrowed’ from somewhere
else the quantity can be denoted negative) is consumed.
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After previous criticism one might consider the model useless. However, this is not the case. A
graph with shorter simulation time reprisenting consentrations of more metabolites versus time is
shown below.
Picture six: Simulation of glycolysis for 1.8 sec using time step of 0.00001 and tolerance 10-7 for 4th
order Runge-Kutta. Note the blue curve for ATP.
The picture shows a rise in glucose concentration. This may be due to some early metabolites
occuring initially as concentrations favoring reaction back to glucose. However, as the pathway
proceeds one can observe correct consumption of glucose. A few important things can be learned
from this picture. First, it would seem that the model represents rather ‘normal’ conditions i.e. not
all glucose is consumed even though the equilibrium favours ATP. The point is, that even when the
glycolytic pathway is functioning in non-starvation conditions it does not automatically convert all
available glucose to pyruvate. Secondly, the curves illustrate nicely both the investment phase of
glycolyisis and the payoff phase. These phases can be spotted by observing how initially ATP is
consumed while ADP concentration rises. About 0.75 seconds from the beginning of the simulation
the ATP level starts to build up. One should also notice that in the end the ATP level is fairly higher
than initially. Third point is that fructose 6-phosphate level increases (orange curve) when more
ATP is produced. As discussed in chapter 2.3. the reaction fructose 6-phosphate + ATP →
fructose 1,6-biphosphate + ADP + H+ which is catalysed by phosphofructokinase (PFK) is the main
control site in the glycolytic pathway. It was also mentioned that ATP is an allosteric inhibitor of
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PFK and as shown in the picture the model actually follows this behaviour: As ATP level increases
the PFK is inhibited, which results in rise of fructose 6-phosphate level.
Simulations with different initial concentrations would be interesting to analyse. Unfortunately, as
stated earlier, the rate constant values are defined for conditions described in table two. Thus if for
example initial concentration of glucose were risen significantly the equation for its conversion to
glucose 6-phosphate would no longer be valid as the allosterically controlled enzyme hexokinase
would then experience rather stronger activation. If one had an acces to kinetic parameters for
different conditions the model constructed could be edited.
5.
Discussion
The exercise consists of three parts: the glycolysis, the Cellware and an integration of these two.
The aim was not only to discuss the modelling of glycolysis with Cellware, which the title of the
exercise suggests. Such a title as ‘Glycolysis, the Cellware program and simulating glycolysis’
might have been more descriptive but is sounds helplessly dull. The exercise was meant to
introduce both glycolysis and the Cellware program for readers not familiar with these concepts. In
addition, the goal was to give an example of systems biology in form of simulation of glycolysis
and also show that in silico methods can give new insight into biological phenomena. Glycolysis
can be studied by reading literature of biochemistry and the Cellware can be learnt from reading the
user manual and using the program. However, the simulation run was unique: the results aquired are
not borrowed, they were created. The latter sounds rather solemn and one surely realizes that the
model described here is just one of the many that can be found on subject. Published scientific
articles include more sophisticated and accurate experiments. Anyway, as the modelling can be
considered as the most important achievement of this exercise the title is quite justified.
Glycolysis is described in chapter two. The text aimed to provide sufficient information for a reader
with little experience in biochemistry to understand the logic behind the model constructed and also
give insight into the results of the simulation. Also as an understanding of the regulation of the
pathway is the key to all kinetic models (or more precisely: defining the kinetic equations and
constants) of glycolysis an emphasis is put on this subject. One may have noticed that chapter two
actually has adopted a rather systems biological point of view: for example, virtually no information
about diseases of malfunctions associating glycolysis is given. Also, even though the structure of
biomolecules is an essential factor to their function, no structures of metabolites are discussed. Are
these features unimportant? Certainly not, but they are rather irrelevant considering the model
applied. After all, the subject was not solely glycolysis.
Chapter three is about Cellware. However, as stated earlier the chapter is not a guide to use the
program. It is shortly introduced and the methods behind its simulation engine concerning model in
chaper four are described. As high school mathematics and chemistry are not sufficient to
understand the function of algorithms or most details of kinetics, chapters 3.2. and 3.3. are not as
elementary as the chapter two. The difficulty is not intentional, rather it is inevitable. If the basics in
solving partial differential equations and physical chemistry were given, the text would be long
enough to cover a few books. Anyway, some of the important properties are derived and one good
reason to do this was that the Cellware manual simply lists these matters but lacks even the
formulas.
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Description of the modelling in chapter four serves as an example of using the Cellware. However
as discussed in this chapter earlier this was not the point. The modelling process including the
gathering of suitable reaction equations and constants prooved to be the most challenging task in the
exercise. Fortunately, the results acquired were satisfactory and the work was not futile. The
meaning of chapter four is analyzed a few times earlier in this chapter. At this point one can self
consider the value of the model but a larger issue is the meaning of in silico research in general.
With better knowledge and more understanding of different sciences better and better models
including large networks of reactions can be constructed. Systems biology can offer methods to
lower costs in medicine development, to get more accurate diagnosis for patients and to provide
new means to optimize different bioprocesses. The integration of older sciences gives birth to a new
branch with potential hard to exaggerate.
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6.
References
[1] J. Berg, J. Tymoczko, L. Stryer: Biochemistry Fifth Edition. W. H. Freeman and Company.
New York. 2002, 423, 436-437, 444, 445
[2] Systems Biology Group, Bioinformatics Institute Singapore: Cellware User Manual, 7, 52
[3] Annals of Biomedical Engineering, Vol 30. M. Lambeth, M. Kushmerick: A Computational
Model for Glycogenolysis in Skeletal Muscle. Biomedical Engineering Society. 2002, 808, 813
[4] Estrella Mountain Community College, Online Biology Book, Mike Farabee, Ph.D.
http://www.emc.maricopa.edu/faculty/farabee/BIOBK/BioBookGlyc.html
[5] Eberly College of Science, department of Biochemistry and Molecular Biology. Dr Sypes:
Lecture notes: Glycolysis reactions
http://www.bmb.psu.edu/courses/bmb211/classnotes/glycolysisrxns.html
[6] J. Clayden, N. Greeves, S. Warren, P. Wothers: Organic Chemistry. Oxford University Press.
Oxford 2001
[7] T. Engel, P. Reid: Thermodynamics, Statistical Thermodynamics & Kinetics. Pearson
Education, Benjamin Cummings. San Francisco 2005
[8] M. Gregoriou, I. Trayer, A. Cornish-Bowden: Biochemistry 20. 1981. 499-506.
[9] Helsinki University of Technology, Laboratory of Mathematics. Timo Eirola:
Osittaisdifferentiaaliyhtälöt, Mat-1.404 Lecture notes spring 2005.
[10] European Journal of Biochemistry. 267, 1-18 (2000). B. Teusink et al. Can yeast glycolysis be
understood in terms of in vitro kinetics of the constituent enzymes? Testing biochemistry
http://www.bio.vu.nl/hwconf/papers/ca20.pdf
[11] The Walter and Elizia Hall Institute. Melbourne. Australia. A Gottschalk: The Mechanism of
Selective Fermentation of d-Fructose from Invert Sugar by Sauternes Yeast (1946). 624
[12] Beilstein-Institut. Frankfurt. Germany. C. Kettner, M. Hicks: Chaos in the world of enzymes How valid is Functional Characterization Without Methodological Experimental data? (2004)
http://www.beilstein-institut.de/escec2003/proceedings/Kettner/kettner.htm
[13] European Journal of Biochemistry, Vol 108, 295-301, (1980). F. Gotz, S. Fischer, K. Schleifer:
Purification and characterisation of an unusually heat-stable and acid/base-stable class I fructose
1,6-bisphosphate aldolase from Staphylococcus aureus
http://content.febsjournal.org/cgi/content/abstract/108/1/295
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