Chapter 8

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 biological system.

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:

 Or

M_TGpalmitate represents the mass (in, say, µmol) 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 P_TGsynthesis and P_lipolysis. 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.

Now 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.

The differential equation for conservation of mass can be written for the tracer molecule just as it was for the total TGpalmitate mass.

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, P_lipolysis.

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 by  yields the equation we need:

Importantly, this equation holds even when all the variables are functions of time and the chemical system is NOT in a steady state.

So, to be completely general, tracer differential equations are related to the chemical system differential equations and can be written as

plus a system of algebraic equations like this

each of which links a tracer flux to the corresponding chemical system flux.

Importantly, if the chemical system is in steady state, all the chemical system fluxes and masses are constant and therefore

which 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:

where k_TGsynthesis and k_lipolysis are rate constants characterizing the processes, P_TGsynthesis and P_lipolysis, respectively.

This 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.

Tracer 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.

In 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 atom’s 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.

But 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.

For 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.

Other 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.

One 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.

But 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 ROI.

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.

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