The Aging of Molecules - a PDE with a Physical Background

When we look at a person we can easily make a good guess how old he or she is by taking into account several signs of aging such as gray hair or wrinkles. This concept of age does not translate directly to biological molecules since neither proteins nor mRNAs get wrinkles, but in both cases we can find different aspects which can be an indicator for a molecule’s age. In the case of proteins it is known that long-lived proteins accumulate damage, mainly caused by reactive oxygen species (ROS). mRNAs however, receive after their transcription a stabilizing element, the poly(A) tail, a chain of about 250 adenosides. This chain is eroded during an mRNA life time. Roughly speaking an “aged” protein is then a protein which accumulated damage and an aged mRNA is an mRNA where the poly(A) tail is shortened. However, both processes, accumulation of damage in proteins and erodation of the poly(A) tail in mRNA, do not always proceed linearly in time as the term “aging” might suggest. For example there might be times during the day with an increased ROS production and consequently a higher protein damage.

Similarly, it has been suggested that erodation or deadenylation of a poly(A) tail happens quite quickly for the first adenosides but slows down for the last ones it. In both cases the “aging” of either proteins or mRNA occurs faster or slower depending on the condition and background.

The McKendrick [241, 242] or von-Foerster [243] equation describes the aging of a population.

I use this model to describe “aging” molecules and modify it in such a way that the molecule’s aging do not occur linearly in time. For simplicity, and to avoid the poor analogy of “aged”

molecules, I will refer from now on to “damaged” molecules. This can be directly translated into damaged proteins. To translate this concept to mRNA with shortened poly(A) tail we calculate the mRNA with a “damaged” tail, a newly synthesized mRNA with a full poly(A) tail has no damage.

In the modified McKendrick model a molecule is characterized by two variables, time t and damaged. It can be represented in a three-dimensional graph, see Figure 3.4A. Let thenx(t, d) be the number of molecules of damage (d, d+ dd) at timet. Hence, x(t, d) is a density function in the damage d. An integration by the whole damage range gives the total concentration of

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molecules X with all possible damages at a specific time t X(t) =

Z ∞ d=0

x(t, d) dd. (3.4)

The concentration of molecules within a certain damage rangeD+∆dis given by an integration over a certain range of damage

X(t, D) =

Z D+∆d D

x(t, d) dd. (3.5)

The fact that the modeling variable actually refers only through integration to a physical property, molecule concentration, requires a careful derivation of the rates. This and a mathe-matical justification of the model is presented in Appendix B.1 and B.2. However, to translate fromx(t, d) toX(t, D) in an intuitive way is rather straightforward, despite the not informative integration. Figure 3.4B illustrates the integration and hence the intuitive understanding for two cases. In short, x(t, d) is linear to X(t, D) for sufficiently small ∆d. Using this linear rela-tionship it is possible to find a translation fromx(t, d) toX(t, D). To avoid cluttered text I will use in the following the term molecule concentration for x(t, d), although technically speaking it is a molecule concentration density.

A molecule which is produced, damaged during its life time and degraded is described with the PDE

∂x(t, d)

∂t =−q(t, d)∂x(t, d)

∂d −x(t, d)∂q(t, d)

∂d −v(t, d)x(t, d).

x(t,0) = k(t) q(t,0)

(3.6)

Here,k(t) is the production rate of newly synthesized proteins without damage, it characterizes the initial condition x(t, d = 0). The rates v(t, d) and q(t, d) refer to molecule degradation and damage accumulation, respectively. Since I want to describe a system under the influence of the circadian clock all rates contain a time dependent, periodic function which describes the oscillating variation of the rates. Degradation and damage accumulation can additionally depend on damage. This can reflect different scenarios: For example, in the case of protein oxidation, highly damaged proteins are more likely to be recognized for degradation than proteins with lower damage. In this case the degradation rate increases with damage. In the same manner the damage accumulation rate could depend on damage as I will show later. The actual mathematical formulation of this damage dependent term depends on the biological situation.

In Appendix B.2 I discuss different scenarios for damage dependent degradation and damage accumulation rates and suggest some mathematical description.

The Equation 3.6, known from population dynamics, is also well known in physics in a very different context. In physics, this equation is used to describe the gas flux through a tube, its name is then “transport equation”. In the following I want to use this physical interpretation to discuss the different influences of the rates. This will, hopefully, provide us with a very intuitive understanding.

In this picture, the concentration of moleculesx(t, d) at timetand damagedis the gas density at time t and position din the tube, see Figure 3.4C. This gas is poured into the tube only at one end of the tube. During this pouring the gas density is varied with a time-dependent rate.

In this way, there are packages of highly dense gas and packages of low density gas produced at the opening and then further transported. In our damaged molecule system, the pouring of gas represents the production ratek(t) of molecules with zero damage. The gas is further transported through the tube with small pumps, with a pump placed on each point along the tube. In this way different speeds of transport can be enforced. Furthermore, all pumps are controlled by one clock which dictates the same periodic rhythm to all pumps. If the small pumps have different power outputs they locally change the gas density. The time dependency of gas transport, however, can not generate a change in gas density since all pumps are controlled together. The time dependency of transport rate only changes the overall transport. Transport changes the position of gas, in the molecule image this is the damage of a molecule, hence the transport reflects the rate of damage accumulation. The last rate to be explained is the degradation rate.

This rate can be imagined as little holes in the tube where gas escapes. Again, the hole size and therefore the escape rate can depend on the position on the tube. But all holes together are controlled by one clock which can close and open the holes. Also this rate is able to change the gas concentration, in this case both time and position dependent. However, degradation rate can only decrease gas density because it removes gas. In contrast to that the transport rate is able to both increase and decrease locally gas density when gas is pooled or thinned out.

It is worth mentioning that the model is only analytically solvable if damage accumulation and degradation rate do not depend on damage, see Appendix B.3. Hence, I will rely on numerical solutions. In Figure 3.5 some numerical simulations for different rates are shown.

This model describing a molecule concentration with two features, time and a property ac-quired over time is very general. Also the two very different examples which I will introduce in the following in more detail underline the model’s generality. In the first example the model is used to describe long-lived proteins and their oxidation, the second describes the poly(A) tail of mRNA.