How to write tracer differential equations
To qualify as a useful tracer a tracer molecule
must be both distinguishable and indistinguishable from the molecule
it traces. It must be distinguishable by some measurement technology
so that, given a sample containing both tracer and traced molecules,
it is possible to measure the tracer molecules. In particular,
we would like a measurement of tracers that is proportional to
the number of tracer molecules in the sample. At the same time
it is essential that the biological system sees no distinction
between tracer and traced. This is the fundamental tracer
assumption: from the perspective of transport, binding
and transformation processes the tracer molecule is indistinguishable
from the traced molecule. It is always your responsibility to
consider the possibility that your particular tracer fails to
meet this test of indistinguishability.
Much has been written and argued about different sorts of tracers
and we will have more to say about this later, but first I want
to suppose that you have a source of useful tracer molecules (that
are both distinguishable and indistinguishable) and you want to
know the equations that govern their kinetics in your particular
This question has been approached many times by both experimentalists
and theoreticians. Here, I want to take what seems to me to be
the most direct route to the final practical result. This is easiest
to do in the context of a specific example and I have chosen the
textbook physiology of adipose tissue. Refer to Figure 8-1 as
you read this brief description of the mechanisms currently thought
to be involved.
If you have already learned the material in previous
chapters, you will recognize several familiar paradigms. All three
of the main classes of biological processes are represented in
this diagram. The Binding class is represented by the interaction
of insulin (in the adipose extracellular space) with its receptor
(INSR), an integral membrane protein of the adipocyte cell membrane.
The Transport class is represented by, among others, the translocation
of the free fatty acid, palmitate, from the adipose extracellular
space to the white adipocyte cytosol mediated by the fatty acid
transporters, FAT and CD36. The Transformation class is represented
by 1) a process that lumps together the reactions of triglyceride
(TG) synthesis from palmitoyl-CoA and glycerol-3-phosphate and
2) the process that lumps together the reactions of lipolysis
catalyzed by the trio of enzymes, ATGL, HSL and MGL. Since you
already know how to write mechanistic rate laws for each of these
classes of processes, and since the derivation of tracer rate
laws does not depend on the mechanistic details, we can start
by writing the differential equation for TGpalmitate in the white
adipocyte lipid droplet:
represents the mass (in, say, Ķmol) or the abundance (in, for
example, molecules/cell) of TGpalmitate in white adipocyte lipid
droplets. P stands for Process or flux and you can see immediately
that this equation is an example or a special case of the fundamental
principle of conservation of mass.† Any change
in TGpalmitate mass depends on the balance between PTGsynthesis
and Plipolysis. If synthesis is greater than lipolysis,
then the mass of TGpalmitate is increasing, which is the same
as saying that its derivative with respect to time is positive.
If, on the other hand, synthesis is less than lipolysis, then
TGpalmitate is decreasing.
suppose that your goal is to write the corresponding differential
equations for a tracer system. Suppose further that your tracer
molecule is a labeled palmitate. This could be a radioactive tracer,
such as 14C-palmitate or a stable isotopic tracer molecule
such as 13C-palmitate.
differential equation for conservation of mass can be written
for the tracer molecule just as it was for the total TGpalmitate
where the * superscript
indicates the tracer mass or tracer flux. The really significant
question is how to calculate a tracer flux, say P*lipolysis,
from the total flux, Plipolysis.
The answer is extremely
simple and derives from the indistinguishability
principle (or assumption) of tracer kinetics: the biological
system does NOT distinguish between a tracer molecule and its
unlabeled analog. More anthropomorphically, the binding, transport
and transformation reactions taking place in the living system
treat a tracer molecule and its native analog exactly the same.
You must always recognize
that the indistinguishability principle is an assumption that may be challenged by other investigators because there
are many ways that a proposed tracer molecule can be a poor or
even an extremely poor tracer. But for the moment we will assume
you have convinced yourself that your tracer molecule is sufficiently
indistinguishable from its unlabeled analog. Quantitatively, the
indistinguishability principle can be written as follows:
This equation asserts
only that the fraction of tracer molecules processed by lipolysis
per minute is the same as the fraction of total† molecules (tracer plus natural) processed by
lipolysis per minute. Multiplying both sides of this equation
†yields the equation we need:
this equation holds even when all the variables are functions
of time and the chemical system is NOT in a steady state.
to be completely general, tracer differential equations are related
to the chemical system differential equations and can be written
plus a system of algebraic equations like
of which links a tracer flux to the corresponding chemical system
if the chemical system is in steady state, all the chemical system
fluxes and masses are constant and therefore
means that tracer differential equations are always linear constant
coefficient ODEs if the underlying chemical system is at steady
state. In this instance, we have:
kTGsynthesis and klipolysis are rate constants
characterizing the processes, PTGsynthesis and Plipolysis,
assertion remains true no matter how nonlinear the underlying
chemical system rate laws may be. It is for this reason that tracer
kinetics has such an extensive mathematical and computational
literature. All the powerful methods of linear analysis are immediately
applicable to tracer kinetics if the chemical system is assumed
to be in steady state.
kinetics as a scientific discipline dates from the dawn of atomic
energy in the 1930s. You need a source of subatomic particles
to synthesize any appreciable quantity of the radioactive isotopes
of atoms that one finds in biological molecules.†
And then, of course, you are faced with a double-edged
sword, namely, you have the radioactive isotope you sought, but
it is decaying exponentially as you begin to make use of it. Not
surprisingly, all the early radio-tracer experiments were done
within a city block of where the radio-tracer was synthesized.
Later, isotopes with relatively long half lives began to dominate
biochemistry, which is why 14C, and 3H,
and 32P, and 35S are familiar to most biologists.
many ways a biological molecule synthesized with radioactive isotopes
in place of the far more abundant stable isotopes, is the perfect
tracer. It is distinguishable because its radioactive decay emits
subatomic particles that can be detected and measured by a variety
of technologies including Geiger counters and liquid scintillation
counters. Sensitivity is excellent and background is usually low.
Moreover, such a tracer is indistinguishable because the chemistry
of an atom depends on its electrons and their energies. Since
the number of electrons is matched to the number of protons in
the atomís nucleus, atoms with the same electronic structure all
have the same atomic number and, consequently all the isotopes
(atoms with the same atomic number) of a given atom will have
the same chemistry.
radioactivity comes with its own set of safety issues and while
any number of scientists and non-scientist volunteers have agreed
to be injected with relatively small doses of radioactively labeled
compounds, radioactivity is no longer administered to human subjects
without some explicit diagnostic or therapeutic purpose.
this reason, the advent of reasonably high throughput mass spectroscopy
and NMR spectroscopy has revitalized tracer kinetic analysis of
human metabolism.† Commercial availability of biomolecules labeled
with isotopes like 2H and 13C, which are
stable but nevertheless rare in nature (only 1.1% of natural carbon
is 13C and only 0.015% of natural hydrogen is 2H),
has enabled an enormous number of extremely informative tracer
kinetic experiments in human volunteers and in patients. A computational
biologist can, today, build an extraordinarily successful career
by offering the tools of ordinary differential equations to experimentalists
in this burgeoning discipline of stable isotope kinetics.
tracer molecules are possible. Indeed, of the last two decades
many new molecular tracers have been introduced. Many of these
are fluorescent, that is, they emit light at a particular wavelength
when irradiated with light of another particular wavelength. The
fluorescent portion of such a tracer (called the fluorophore)
is almost always an aromatic ring structure which may be conjugated
to the traced molecule using any number of chemistries.
class of fluorescent tracers deserves special mention. These are
the fluorescent proteins. Typical of these is the green fluorescent
protein first isolated by Osamu Shimomura†
from a jellyfish native to the Puget Sound. Subsequently
this protein was first expressed as a transgene by Martin Chalfie.
Subsequently, the enormous experimental power to be derived from
a fluorescent protein was then realized and exploited by Roger
Tsien who developed and commercialized this technology by synthesizing
an enormous variety of fluorescent proteins permitting output
signals over the entire visible spectrum.
the real significance of a fluorescent protein is that every protein
is coded for by a specific DNA sequence. Consequently, by applying
the tools of modern molecular biology, it is a relatively simple
matter to synthesize a genetic construct, or transgene, that encodes
your favorite protein fused to the DNA sequence that encodes for
one or another fluorescent protein. This fusion protein will be synthesized by any cell that can be successfully
transfected with the transgene. The beauty of this is that if
you then place your transfected cells on the stage of a suitably
equipped fluorescence microscope, you can image with the requisite
excitation wavelength and visualize
your protein wherever it is localized in the cell. Moreover, it
is often possible to initiate transients and record movies (dynamics!)
as your fluorescent protein moves from place to place in the cell.
Shimomura, Chalfie and Tsien were awarded the 2008 Nobel Prize
in Chemistry for the discovery, expression and development of
Green Fluorescent Protein (GFP).
FRAPs,† iFRAPs and FLIPs
Tracers can be incredibly useful probes of biological systems, but fluorescent
tracers have unique properties that render them especially powerful.
Foremost among these properties is that fluorescent tracers can
be photobleached. Photobleaching typically involves irradiating
a selected region of interest (ROI) with a higher intensity, short
duration laser pulse sufficient to render the fluorescent protein
fluorophore unresponsive to subsequent excitation evan at the
normal input wavelength.
FRAP (Fluorescence Recovery After Photobleach) (some insist on
Redistribution instead of Recovery) involves bleaching a particular
ROI and then measuring the recovery of fluorescence in the bleached
iFRAP (inverse FRAP) involves bleaching an ROI and then measuring
the decrease in fluorescence in some OTHER ROI.
FLIP (Fluorescence Loss In Photobleaching) involves continuous
bleaching in some ROI and measuring the decrease in fluorescence
in some other ROI.
Importantly, photobleaching can be included in tracer differential
equations simply by subtracting a term proportional to the current
fluorescence. This "mass action" term results in an
exponential decrease in fluorescence in the bleached ROI, and
the constant of proportionality is dependent on the intensity
of the bleaching laser beam.
To apply a FRAP, iFRAP, or FLIP protocol to your model, you need
only include these exponential terms in each differential equation
for a tracer state that is present in the bleached region. Notice,
however, that these terms must not be included in the chemical
system equations since photobleaching has no effect on the biochemistry,
binding, or transport of the tagged molecule.
Just to quickly emphasize the dramatic versatility of fluorescent
proteins, it is important to know that there are also photoactivatable
GFPs. These tracers have the intriguing and often very useful
property that they can be turned ON by an input laser pulse. Thus,
a photoactivatable GFP is NOT fluorescent when synthesized in
the cell, but BECOMES fluorescent when irradiated at the appropriate
wavelength. A complete tracer modeling system must be capable
of handling these increasingly common photoactivation experiments.