Chapter 4
Translating Your System Diagram to Mathematics
Conservation of mass is the principle that underlies
translation of your diagram to the language of mathematics. For
each symbol in your diagram, you will write a differential
equation, and the terms on the right hand side of the
differential equation will correspond to the solid arrows that
originate or terminate on that symbol. Consider the example of a
specific messenger RNA (mRNA).
Recalling that a compartment is a chemical species in a
physical place, you want to answer the question: What are the
determinants of the rate of change of mRNA (the chemical species)
in the cytosol (the physical place)? The answer, as adumbrated in
the previous chapter, is that all the processes (arrows)
delivering mRNA to the cytosol or removing mRNA from the cytosol
must be represented by terms on the right hand side of the
differential equation you write.
where m is the mass of mRNA in the cytosol, F_{npc} is
the flux of this particular mRNA through the nuclear pore
complex, F_{nuc} is the flux of mRNA degradation to
nucleotides by cytosolic nucleases, F_{bind} is the flux
of mRNA binding to ribosomes, and F_{unbind} is the flux
of mRNA being released from ribosomes. This example was chosen
because it includes all three types of biological processes:
translocation of mRNA from the nucleus to the cytosol,
transformation of mRNA to individual nucleotides by the action of
a cytoplasmic nuclease, and binding of mRNA to cytoplasmic
ribosomes. The differential equation is the mathematical
statement of conservation of mass for this particular mRNA.
The flux terms on the right hand side of the equation can be
considered as inputs and outputs. These fluxes are thus the
determinants of all changes in cytosolic mRNA mass. Any change in
one of these fluxes must result in at least a transient change in
mRNA mass. As written here, the derivative on the left hand side
of the differential equation has dimensions of mass per unit
time. This means, of course, that the flux terms also have
dimensions of mass per unit time.
Among biochemists, such differential equations are commonly
written in terms of concentration. This makes good
physicalchemical sense. Unfortunately, writing differential
equations in terms of concentration produces several practical
problems when building kinetic models of living systems. The most
obvious of these is that the underlying principle is conservation
of mass, not conservation of concentration. To see this
difficulty more clearly, calculate what happens to the nuclear
and cytosolic mRNA concentration if 20 copies of the mRNA are
translocated from the nucleus to the cytoplasm. To make this
comparison it is essential to know the relative volumes of the
two cellular compartments; so if we make the reasonable
approximation that cytosolic volume is 1 pl and nuclear volume is
0.1 pl, the change in nuclear mRNA concentration is 200 copies
per pl while the change in cytosolic mRNA concentration is only
+20 copies per pl. Our conclusion must be that whenever your
system consists of two or more compartments with different
volumes, it is prudent to write the differential equations in
terms of mass, rather than concentration. It is, of course,
possible to use concentration state variables, but your equations
will be cluttered with a variety of volume ratios whose net
effect is to recast the equations in mass terms.
Biological experiments are conventionally begun at a time when
nothing in the system is changing. This is the condition of steady
state or stationary state. To state this condition in precise
terms, it is a state characterized by constancy of all the state
variables, x_{i}:
where the inverted A means "for all".
Steady state is the essence of life. If a cell has the means to
maintain a steady state, it lives; if not, it dies. It is
critical to realize that the steady state does not imply or
require that the fluxes are all zero. In the case of the mRNA
system, for example, steady state is achieved whenever
that is, whenever the inputs balance the outputs. Rarely do we
have the means to prove that our biological system is in steady
state. We simply cannot measure all of the state variables.
Despite this obstacle, the steady state assumption is a nearly
universal feature of modern biological investigation. Few
scientists, however, can distinguish the concept of steady state
from the more restrictive concept of equilibrium. If you are
interested in the construction of biological kinetic models, you
should be expert about this distinction.
Steady State Is Not Equilibrium
At equilibrium all chemical potential gradients are zero.
Moreover, the principle of microscopic reversibility guarantees
that, at equilibrium, the rate at which any process proceeds in
the "forward" direction is exactly balanced by the rate
of that process in the "reverse" direction. This holds
for every individual process, and means that at equilibrium there
can be no net flux through any enzyme, any ion channel, or any
transport protein. This is easy to understand intellectually
because a nonzero chemical potential difference is absolutely
required to drive a net flux. But steady state and equilibrium
are so commonly used as if they were synonyms, that you will
likely have to think about these points for hours before you can
parry the objections of scientists who are sure of their
misinformation.
A widely misunderstood example is provided by the transport of
ions across the plasma membrane of a cell. Take Na^{+} as
an example. Because, in the normal state of the cell, there is no
net flux of Na^{+} across the cell membrane, and because
the Na concentration in the cytosol is not changing with time,
this condition is sometimes thought to be an equilibrium state.
It is not. It is a steady state. The fact that [Na^{+}]
is constant is sufficient to define a steady state, but is
insufficient to distinguish a steady state from an equilibrium.
That is because an equilibrium is a special case of a steady
state; an equilibrium is a steady state that is achieved when all
chemical potential gradients have decayed to zero and there are
no further net movements of molecules via any process. This is in
marked contrast to the steady state of Na in a living cell. Here,
there is a substantial net flux of Na into the cell through Na
channels in the cell membrane, and there is an opposite but equal
net flux of Na extruded from the cell by the action of the Na^{+}K^{+}ATPase.
There is a nonzero chemical potential gradient consisting of both
chemical and electrical terms that propells Na^{+} into
the cell, and there is a nonzero chemical potential gradient
including a term for the hydrolysis of ATP that pumps Na^{+}
from the cell. Neither the ion channels nor the Na pump can
qualify as processes at equilibrium; there are net Na fluxes
through both. The fact that there is no net flux across the
membrane is simply a corrollary of the steady state, not an
indication of equilibrium. This is because the equilibrium
condition is a statement about processes, not about state
variables. Consequently, equilibrium is attained only when there
are no net fluxes through ion channels and no net fluxes through
the pumps. Equilibrium is thus achieved only when the cell is
dead. Far from being synonyms, the difference between steady
state and equilibrium is the difference between life and death.
Translocation and transformation processes are rarely at
equilibrium in living systems, but binding processes often are.
If we accept, for the moment, the oversimplification that mRNA
binding to ribosomes is typical of other binding processes, then
a quantitative description of this process could be based on
classical binding theory and the resulting equations could be
used for a wide range of ligandreceptor interactions.
The equilibrium constant for a binding reaction is called the
dissociation constant if the numerator of the equilibrium
concentration ratio is chosen as the product of the unbound
ligand and unbound receptor concentrations. In this case the
denominator is the concentration of the ligand receptor complex.
where K_{D} is the dissociation constant, [m] is the
equilibrium concentration of unbound mRNA, r is unbound
ribosomes, and mr represents the mRNAribosome complex. The units
of a dissociation constant are concentration. If the binding
process is viewed from the other side of the reaction, the
equilibrium constant is referred to as an association constant
and the corresponding equation is
where K_{A} is the association constant (units of
concentration^{1} ) and the other symbols are as before.
A simplification, characteristic of biological systems, is that
one of the reactants, usually the "receptor", is
limiting. As a consequence, the binding process saturates when
there are no unbound receptors remaining. If the total amount of
"receptor" (in this case, ribosomes) is given the
symbol, B_{tot}, then conservation requires that B_{tot}=[mr]+[r]
or [r]= B_{tot}[mr]. Substituting this for [r] in the
expression for KD and solving for [mr] yields the classical
binding expression:
which gives the amount of bound mRNA as a function of the
"free", or unbound, mRNA concentration.
This is such a compact description of binding, that modelers are
often tempted to use this equation to represent binding processes
which they believe to be in "rapid equilibrium". The
assertion of "rapid equilibrium" has to be evaluated in
two parts. We have already explored what is meant by
"equilibrium", so the question is what additional
assumption is conveyed by adding the word, "rapid."
What is usually meant is that the processes represented by F_{bind}
and F_{unbind} are so fast that reasonable transients in
the concentration of ligand never displace the binding reaction
from it's equilibrium point. In terms of our example, this means
that a sudden increase in transcription of the gene that codes
for our mRNA, and the resulting increase in translocation of this
mRNA from the nucleus to the cytoplasm will cause an increase in
[m] that will result instantaneously in an increase in [mr] that
always keeps the ratio of [m][r]/[mr] constant and equal to the K_{D}.
Thus, the system of equations that would be written is
Notice that the two fluxes, F_{bind} and F_{unbind},
are missing from the differential equation. Instead, the binding
process is embodied in the algebraic binding equation, and these
equations can be solved simultaneously to give m(t) and mr(t).
Unfortunately, these equations fail to enforce conservation of
mass on the dynamic system. You can see this by imagining the
consequences of a sudden increase in the number of ribosomes,
that is an increase in B_{tot}. The calculated amount of
bound mRNA, [mr], will increase immediately since [mr] is
proportional to B_{tot}. But where does this increase in
bound mRNA come from? The disturbing answer is that the increase
does not come at the expense of free mRNA. In fact, the increase
arises from mathematical "thin air." Because there is
no term in the differential equation for m that represents loss
of mRNA to the bound state, the equations are "unaware"
that free m must decline. Clearly, this violates conservation of
mass.
There are two methods for solving this problem. The first is
to write out the complete system of differential equations
including the F_{bind} and F_{unbind} terms. This
enforces conservation of mass automatically, but can lead to
stiff differential equations if the binding and unbinding
processes are much faster than the other processes in the system.
We will have more to say about stiff systems later.
The second is to write the differential equation for total
mRNA (call it M) in the cytosol rather than for free mRNA in the
cytosol. The equations are then
In effect, this method uses the differential equation to keep
track of the total amount of mRNA, and uses the equilibrium
binding equation to distribute this total between the bound and
unbound forms. The two algebraic equations can be solved
simultaneously for m and mr, but the result is a fairly involved
quadratic of the kind that professors always seem to leave as an
exercise for the student.
I got
and I would love to be corrected or corroborated by my students.
Experience suggests that, whenever possible, you should write out
and solve the complete system of differential equations.
Numerical methods for the solution systems of nonlinear
differential equations are much more efficient and robust than
methods for the solution of systems of nonlinear algebraic
equations.
Updating Your Assumption List
If you assume that your biological system is in steady state
during all or part of your experiment, a statement to this effect
should be added to your assumption list. If you assume that a
binding process is at equilibrium throughout your experiment,
this should be made explicit as well.
Translation Versus Interpretation
Your job as a kineticist is to produce a system of equations
that is a faithful reproduction of the hypothesis to be tested.
Often, the hypothesis will be stated as a series of sentences or
paragraphs filled with the imprecision that enlivens fiction,
allows a poem to mean different things to different readers, and
is a universal feature of human language. The process of
translating the stated hypothesis to mathematics is the heart of
our endeavor. If the hypothesis is your own, you will almost
surely find that the act of converting it to mathematics forces
you to think more deeply about your system and perhaps even alter
the hypothesis long before you begin to test it against the
experimental data. If the hypothesis was formulated by someone
else, you will almost surely find that you want to talk with him
or her. Over and over again, you will want clarification of some
detail of the hypothesis. Often, you will find that implicit
assumptions become explicit (and get added to your assumption
list) as you progress. Ultimately, however, the mathematical
model is your best guess about how to translate the stated theory
to mathematics. The more you understand about mathematics,
chemistry, physical chemistry, kinetics, and biology, the better
will be your best guess. To be a good biokineticist, you need
expert knowledge of both biological science and physical science.
The process of translating a working hypothesis to mathematics
is, perhaps, better seen as interpretation. The process of
translation can be carried out by anyone with a language_{1}tolanguage_{2}
dictionary, but the result is nearly worthless. Interpretation,
as it is done by the unseen experts at the United Nations General
Assembly, requires that the interpreter be expert in the idioms
and culture of both the speaker and the listener. Your job is
equivalent; you must be expert in the idioms and culture of both
physical chemistry and biology.
Recognizing this as the ideal situation, we can embark on a
journey that teaches this wisdom by the method of John Dewey. His
maxim: "We learn what we do."
A StepbyStep Procedure for Translation
 Define Your System Boundary in Quantitative Terms
 Formulate Your Rate Laws
 Add Regulation and Control to Your Rate Laws
 Link Your State Variables to Experimental Measurements
Step 1: Define Your System Boundary in Quantitative Terms
In reality, no system variable is ever constant, but for
practical purposes many system variables may be treated as
constant during the experimental protocols you choose to impose.
Constancy, whether measured or assumed, constrains your model and
helps define the boundary of your system.
Some common examples are: constant
 temperature
 pressure
 pH
 osmolality
 enzyme concentration
 PO2
 volume
These features of the system do not define the physical
boundary of the system, but they are part of the boundary
nonetheless. This is because they define parts of the system for
which you will not write differential equations. Sometimes the
actual value of the constant is known. If so, you convert the
value to consistent units and enter the result in the appropriate
part of your model file. If the value is not known, you add it to
the parameter list if you expect the experimental data to contain
information on its value, or you assume a value and add this
assumption to your Assumption List.
When a system variable has a known time course during your
experimental protocol, it is usually used as a constraint on
model structure by requiring that the model's differential
equations yield the observed time course. But if the variable
whose time course is known lies on the boundary of your model,
then we frequently impose the constraint by using a forcing
function. In the context of kinetic modeling, a forcing function
is either a time series of data points or an algebraic function
of time that serves as a lookup table supplying the variable's
value whenever the model solution requires it. When a series of
data points is used, they are usually connected by linear
interpolation, so that values of the variable are available to
the numerical integrator at times when no observation was made.
Alternatively, the data points may be fitted to an algebraic
function, such as a power series in t, and this function can
supply the required values at any required time, t.
Occasionally, forcing functions are used in place of system
variables whose differential equations are part of the model.
This is done to decouple the system so that related parts of an
unfinished model can be developed in parallel. The advantage of
using forcing functions is that each part of the model is
developed using "correct" (as defined by the
experimental data) inputs, rather than the "incorrect"
inputs that result from solving the differential equations of a
model still under development. As the modeling process converges
on a consistent model, the forcing functions are removed, and the
model's differential equations take over as suppliers of inputs
to the rest of the system.
In summary, the boundary of your dynamic model is defined by
specifying those system variables for which you will not write
differential equations. This boundary is quantified by specifying
the constant numerical values of these variables, or by providing
an explicit function of time to be used by the model whenever a
value for the variable is required.
Step 2: Formulate Your Rate Laws
Rate laws are the heart of kinetic modeling. A rate law is an
algebraic expression which can be evaluated to give the flux of a
given chemical species through a given process. The term flux is
used here to mean the mass (of the chemical species) per unit
time passing or moving through the process. Typical units of flux
are fmol/sec or umol/min. Since most biological processes are
mediated by proteins, most rate laws are quantitative statements
of the function of one or another protein. In some biological
disciplines, the word flux is reserved for a normalized quantity,
namely, the mass per unit time per unit crosssectional area.
This is especially useful in studies of membrane transport. In
this case, typical units are pmol min^{1}cm^{2}.
Writing rate laws requires careful application of physics,
chemistry, and scientific imagination. A fundamental tenet of
chemical kinetics is that the probability that a process proceeds
or that a reaction takes place depends on the frequency of
molecular collision of the reactants. Moreover, collision
frequency is proportional to the product of the concentrations of
the reactants since the probability that a molecule of each
reactant is present in a given microvolume increases with its
concentration in the bulk solution.
Hint concerning units: In solution, of course, concentration is
measured as mass per unit volume. Typical units are pmol/liter or
mmol/liter. SI units no longer recognize the simpler, but
frequently misused, concept of Molar. Nevertheless, the symbol,
M, frequently appears in the biological literature and means 1
mole/liter. Thus, 1 pmol/liter = 1 pM. The symbol, M, must never
be used to signify moles. A concentration given as 1 pM/liter is
nonsense.
Concentration, however, is not the only physical determinant of
collisions. If two solutions have the same concentrations of the
reactants, it is still possible for one of these solutions to
produce many more collisions per unit time than the other. This
could happen for two reasons; one simple, one not. First, the
simple reason: there might be more of one solution than the
other; more collisions will take place because there are more
molecules of both reactants available. This approach to
increasing the flux through a given process is adopted in
essentially all multicellular organisms. By synthesizing a
molecule, say insulin, in many pancreatic beta cells rather than
in only one, the organism can produce insulin at a rate
proportional to the number of working beta cells as well as
proportional to the concentrations of the required substrates.
A less obvious means of increasing collision frequency (and
therefore flux), without increasing concentration, is to increase
the kinetic energy of the reactants by increasing the
temperature. This works is two ways: 1) increasing the
temperature increases the number of collisions that occur per
unit time because the reactants travel at greater velocities
between collisions so the next collision will occur sooner, and
2) increasing the temperature increases the number of collisions
between reactants with energies sufficient to drive the reaction.
These collisions may be referred to as successful collisions.
Combining these ideas yields the fundamental rate law:
where Flux is the mass of product formed per unit time (mol sec^{1}
), A and B are the concentrations (mol/liter) of the two
reactants, f_{c}(T) is the temperature dependence (sec^{1}
M^{1} )of the frequency of collision, S is the
probability (unitless) that the spatial arrangement of the
reactants at the moment of collision is appropriate for
initiating the reaction, F_{a}(T) is the fraction
(unitless) of the reactant molecules having sufficient energy to
yield a successful collision, and V is the volume (liters) in
which the reaction takes place.
Temperature increases f_{c} only linearly so a 10^{o}
change will typically alter f_{c} by less than 5%, but
the temperature dependence of F_{a} is much more
dramatic. You can see this by starting with the concept of
activation energy. The activation energy, E_{a}, of the
reaction is the minimum free energy the reactants must have to
initiate a successful collision. We know from the theoretical
work of Maxwell and Boltzman that the fraction of molecules
having energy greater than or equal to E_{a} is an
exponential function of temperature, namely
This means that the flux will approximately double, for an
activation energy in the usual range of 6000 to 15,000 cal/mol (6
to 15 kcal/mol), if the temperature is increased by 10^{o}.
Since values of RT are about 0.54 kcal/mol at 0 ^{o}C,
0.58 kcal/mol at 20 ^{o}C , and 0.62 kcal/mol at 37 ^{o}C,
values of F_{a} are very small numbers in the range of 10^{8}
to 10^{7}. Nevertheless, because of the exponential
dependence, Fa changes by significant factors over this small
range of RT. Consequently, you must expect changes in reaction
rates when temperature changes. If, on the other hand,
temperature is constant, the flux equation becomes
where k(T) is the second order rate constant of the reaction, and
is itself frequently written as the Arrhenius equation:
formulated empirically in 1889, well before the development of
the collision theory by Eyring in 1935. Here, K_{A} is
the Arrhenius constant that lumps together all the factors that
either do not depend on temperature or depend on it only weakly.
Exercise to get a feel for the magnitude of K_{A}:
You have found that the rate constant for a given reaction is 0.1
sec^{1}. What is the value of K_{A} if the
activation energy of the reaction is 10 kcal/mol and the
temperature is 20 ^{o}C?
Your choice of rate law is determined, in part, by your
knowledge (or assumptions) concerning which of the chemical
species involved are changing with time. Indeed, if you believe
that one participating species is present in such large
quantities, that its concentration can be assumed constant, then
this species becomes part of the boundary of your system, and its
constancy becomes one of your assumptions.
Absolute values and choosing the units for your States
Units have fascinated and tortured me for 44 years. From freshman physics at MIT where we amused
ourselves calculating the speed of light in furlongs per fortnight (life was
simpler then), to the present day when I’m wordlessly tormented by famous biologists
who blithely speak of moles and molar as if they were synonyms or, worse, by
experimentalists who have no interest at all in real absolute values of the
things they measure. “Oh, that’s not important,” they say. “We’re only
interested in differences here.”
Years ago, in a fit of cynicism, I gave up complaining about axes labeled “percent of control” or
“foldincrease” or even “amplitude.” But having become part of the system, a
card carrying biologist, I find I cannot resist fighting a sort of guerrilla
war to unseat normalized values and put units on their rightful throne. I love
biology, and I am a biologist, but I’m quite sure we cannot reach the goal of
understanding human biology well enough to solve complex diseases without
understanding and using units well.
Engineers and physicists and chemists do not need convincing. They already wince when reminded of a multi
million dollar mission to Mars that was lost because a single number was not
converted from English to metric units. If you have read this far, you must have
begun to suspect that biology is too complex to be comprehended without
mathematical and computational help. But if you finish this book and go back to
publishing “percent of control” graphs without at least thinking to yourself,
“It would be better to have absolute values here,” I will have failed in an
important respect.
What do I mean by an “absolute value?” I mean a number with units and a numerical value I can trust.
It doesn’t have to have 15 significant figures, like the speed of light in a
vacuum or the gas constant. In fact, if it’s accurate to within 15 or 20
percent, I’ll be endlessly grateful, but we biologists have forgone accuracy
and seem to worship at the altar of precision or reproducibility instead.
Everyone aims for reproducibility. Exquisitely tight error bars abound. But only transplanted
physicists and engineers seem willing to invest the time, and it can take a lot
of time, to make a measurement that is not merely proportional to the quantity of interest,
but actually is the quantity of interest.
Obviously, if you only have a measurement that is merely proportional
to the real value, normalizing or forming a ratio is an excellent strategy. The
unknown constant of proportionality disappears in the magic of long division.
and you reap the added benefit that even if you had no idea what the correct units
were, they have become immaterial since the ratio is
both unitless and beyond criticism. No wonder people like ratios!
In other words, reproducibility is no guarantee of accuracy. A wonderfully reproducible
measurement can be inaccurate by orders of magnitude if the person making the
measurement makes the same error every time.
There are several ways you can convince yourself that this is a common problem. First, go to your favorite
biological journal and look at the vertical axes of 100 graphs in 100 different
published papers. Somewhere between 70 and 90 of those graphs will plot a
normalized variable or a ratio of experimental to control. Absolute values are
rare; ratios are rampant.
A second way to convince yourself that this is a serious issue is to search through the biological
literature for the intracellular concentration or abundance of your favorite
protein. The web search is straightforward: type your protein’s name and
“molecules per cell” in the search box.
What you will find, I think, is that reported values (even in the same cell type) commonly differ by factors
of 5 and sometimes differ by orders of magnitude. Can you imagine any other
human endeavor where the available measurements of important variables differ
to this extent and yet do not provoke serious doubts about methodology?
There are individual investigators and even entire subfields of biomedical research for whom
accuracy is a high priority, but many of us, recognizing that it’s harder and
costlier to make absolute measurements, choose the easier and widely sanctioned
course. We normalize measurements to their control values or their peak values
or to a “housekeeping” gene, or to some other measurement that makes every subsequent
measurement relative.
Problems caused by arbitrary scaling of states
“Why not?” you might ask.
What is really wrong with this approach?
There are two important problems: one theoretical and one practical.
The theoretical problem is that whenever you make two measurements and then report only the ratio of the
two, you lose information. Suppose you are the GM of a major league baseball
team. You’re considering two minor league prospects; player A has a batting
average of .333 and player B a batting average of .310. Would you really want
to make your decision without knowing that A had only 3 atbats and B had 500?
A genie emerges from your greatgreat grandfather’s oil lamp and offers you one
of two rewards for setting him free: one is 50% gold and the other is 10% gold.
Would you really want to choose without knowing that the first is a ring and
the second is a castle?
The practical problem is more subtle, but its implications are profound, especially for modelers of
biological systems. It arises when a modeler decides to express all his/her
system variables as normalized or scaled quantities.
Chemical engineering has a long and distinguished tradition of expressing its equations in unitless scaled
variables. This has always been attractive to mathematicians and software
engineers for whom the speed of light is “c.” They have little interest in
whether your speed of light is in meters per second or furlongs per fortnight.
They simply, and reasonably, assume you will not presume to apply their
formulas without using consistent units for all the quantities involved.
Personally, I agree that nondimensionalized equations are more general, but I
find them far less intuitive, so I don’t recommend them. But the success of
nondimensionalization hinges on a very careful procedure, and I fear that the
prominence of this approach has led to the potentially dangerous practice of
arbitrary normalization of state variables.
What do I mean by dangerous? Let’s consider a simple biological subsystem:
Shown here is a simple prototype biochemical reaction regulated by a modifier molecule. The modifier
is itself regulated by binding to a partner molecule forming the modifier:binding partner complex.
What I want you to think about is that choosing to normalize state variables in an arbitrary way, such
as “all concentration variables are scaled to be dimensionless numbers of order
1,” can have undesirable consequences. In particular, if the molecular
abundances of the modifier and its binding partner are very different, the
ability of the modifier to control conversion of reactant to product will
depend strongly on whether modifier >> (is much greater than) binding
partner or binding partner >> modifier. The problem with scaling the
state variables is that this potentially crucial dependence on relative
abundance can never be observed.
You can test this for yourself. Create this model in your favorite ODE solver or use the public
model, “modulator binding” in the ProcessDB database.
Create a model realization of your own and specify the following initial
conditions:
modifier = 100 molecules/cell
reactant = 10,000 molecules/cell
binding partner = 100,000 molecules/cell
modifier:binding partner complex = 0
product = 0
second order rate constant for modifier binding = 1E5
rate constant for modifier:binding partner unbinding = 0.2
A simple rate law for the reaction will suffice:
reaction = k*reactant*modifier
k = 0.01
Solve this model. Save the solutions for modifier(t) and reaction(t). Then change
the initial conditions so that
modifier = 100,000
binding partner = 100
This is just the opposite of the first solution. Solve it and compare the reaction(t)
solution with the saved solution. The difference between solutions should leave
no doubt in your mind that the model is very sensitive to modifier and binding
partner abundances.
Now the next part of the argument aims to convince you that arbitrary scaling will lead to the wrong
answer. You can test this idea either by simulation or by just thinking. Here’s
the thinking approach.
If the abundances of modifier and binding partner were scaled so that each was normalized to its
initial value, then both initial conditions would be 1.0. I imagine it will be
obvious to you that interchanging these scaled initial conditions will cause
absolutely no change in the simulation results for reaction(t).
Consequently, arbitrary scaling of state variables cannot be recommended because there are many
situations where it will yield the wrong answer.
One further biological example may help solidify this concept if you have ever made up a buffer.
Buffers are basically weak acids with equilibrium constants near the pH you
want to maintain. But if you make up your buffer with 1 nM HEPES instead of 10
mM HEPES, your buffer strength is totally inadequate and your solution is at
risk for wild swings in pH. If you substitute “H^{+}” for modifier and
“weak acid” for binding partner, then the above diagram would represent a
pHcontrolled reaction in a buffered solution. Obviously, if protons are far in
excess of weak acid, the reaction will in all likelihood stop dead. Relative concentrations
or relative abundances are important; arbitrary scaling should not be employed
without careful consideration of these issues.
A recommendation on units
Here, then, is a simple recommendation on units for States that will serve you well. For cellular
systems where you are primarily or exclusively interested in intracellular
processes, consider using molecular abundance (expressed in molecules/cell or
copies per cell) instead of any volume based measurement of concentration such
as pmol/liter or µM. For extracellular States it is often easier to use
concentration units, but remember that you will want to be sure that all your
rate laws yield fluxes in molecules/sec or molecules per some other unit of
time or pmol/sec, especially if any of your processes are transport processes
that move molecules from one physiological place to another.
Abundance is particularly useful when some of the model’s molecular species are (or are bound to)
integral membrane proteins or lipids in the membrane bilayer. Some modelers
will insist that the only acceptable measure of chemical activity for use in
rate laws is concentration, but this is probably a historical bias based on
chemical kinetics in a spectrophotometer cuvette.
Consider two rate laws for the same second order process:
The first expresses the flux, P_{1}, as a function of the molar concentrations of the reactants, [S_{1}] and [S_{2}].
The second expresses the same flux as a function of the abundances S_{1}
and S_{2}. Both are correct, so they must be equivalent. What must be
true for this equivalence to be true? First we can express the concentrations
as
where S_{1} and S_{2} are the abundances
of the two reactants in units of molecules/cell, V_{1} and V_{2}
are the appropriate volumes of distribution (in units of liters per cell) for
the intracellular compartments where these molecules are distributed and N_{A}
is Avagodro’s number, 6E23 molecules/mole.
So all that is required for equivalence of the two rate laws is that
In other words, the only difference between using concentration and using abundance is the value of the
rate constant. In return for this unfamiliar rate constant you avoid the
nagging problem of never knowing the appropriate volumes of distribution for
membrane proteins and lipids. By using abundances, you reduce the number of
unknown parameters in your model and improve your quality of life.
Next, we consider how to write rate laws for the major classes
of processes occurring in biological systems: binding,
transformation and translocation.
