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, Fnpc is
the flux of this particular mRNA through the nuclear pore
complex, Fnuc is the flux of mRNA degradation to
nucleotides by cytosolic nucleases, Fbind is the flux
of mRNA binding to ribosomes, and Funbind 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 physical-chemical 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, xi:
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.
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 ligand-receptor 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 KD is the dissociation constant, [m] is the
equilibrium concentration of unbound mRNA, r is unbound
ribosomes, and mr represents the mRNA-ribosome 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 KA 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, Btot, then conservation requires that Btot=[mr]+[r]
or [r]= Btot-[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 Fbind
and Funbind 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 KD.
Thus, the system of equations that would be written is
Notice that the two fluxes, Fbind and Funbind, 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 Btot. The calculated amount of bound mRNA, [mr], will increase immediately since [mr] is proportional to Btot. 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 Fbind and Funbind 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.
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.
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 bio-kineticist, 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 language1-to-language2
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."
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
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 look-up 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.
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 cross-sectional area.
This is especially useful in studies of membrane transport. In
this case, typical units are pmol min-1cm-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, fc(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, Fa(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 fc only linearly so a 10o
change will typically alter fc by less than 5%, but
the temperature dependence of Fa is much more
dramatic. You can see this by starting with the concept of
activation energy. The activation energy, Ea, 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 Ea 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 10o.
Since values of RT are about 0.54 kcal/mol at 0 oC,
0.58 kcal/mol at 20 oC , and 0.62 kcal/mol at 37 oC,
values of Fa 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, KA 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 KA: You have found that the rate constant for a given reaction is 0.1 sec-1. What is the value of KA if the activation energy of the reaction is 10 kcal/mol and the temperature is 20 oC?
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.
Next, we consider how to write rate laws for the major classes of processes occurring in biological systems: binding, transformation and translocation.
You may return to theTable of Contentsor proceed to Chapter 5 .