How are Saturating Removal (SR, Uri Alon et al.) and Minimal Aging model (Fedichev & Gruber) related?

Let's first remind ourselves of the definition of both aging models.

Note: Parameters that appear in both models with the same letter, will have a suffix $\text{SR}$or $\text{FG}$to indicate their origin.

Both models represent a differential equation for a particle in a potential well, undergoing stochastic random noise and are a Langevin equation.

In a generic sense, we can write

\begin{equation} \label{eq:generic_langevin} \frac{d x}{d t} = A t + B - C x + G x² + \sqrt{2 D} ξ(t), \end{equation}

where we have a direct time dependence with coefficient $A$, a constant coefficient $B$, a negative linear term in $x$, a quadratic term in $x^2$and the stochastic random noise term with coefficient $\sqrt{2D}$, where I have written the time dependence of the random process explicitly. The latter is what makes this a stochastic differential equation (and the omission of the explicit time dependence in both papers is imo annoying).

When comparing the two models above with this equation, we can see that they sort of follow this equation, but not quite:

• the SR model does not seem to have the terms $-Cx+G{x}^2$,
• the F&G model has a term $\beta \text{'}{z}_0\gamma t$in addition, which would correspond to something like $Ttx$in our notation here (with a new parameter $T$).

In the next few sections we will show that these differences are to a large extent cosmetic.

The form presented in eq. \eqref{eq:generic_langevin} does not seem to show up in the SR model. The equivalent of the $x$variable in the SR model is the $X$variable. $-Cx+G{x}^2$contains all terms that are proportional to $x$in some sense. Thus, the corresponding term in SR, is

\[ -β_{\text{SR}}\frac{X}{X + κ}. \]

Now, if you are a physicist like me or someone with more applied math experience, you will probably realize that these two are more similar than they appear at a first glance. Consider this:

• at very small $X$values where $X\ll \kappa$, the fraction behaves like a linear (!) function
• at slightly larger values the value of $\kappa$becomes more important. This slows the growth of the linear curve. This can be modeled by a negative quadratic term.

To make this more concrete, we can do the Taylor expansion ¹ of this fraction around $X=0$,

\[ \frac{X}{X + κ} = \frac{X}{κ} - \frac{X²}{κ²} + \frac{X³}{κ³} - \frac{X⁴}{κ⁴} + \mathcal{O}(X⁵). \]

If we ignore all terms of $X^3$or higher, this is exactly the form of the generic Langevin equation.

Applying this to the SR model as a whole yields the following,

\begin{equation} \label{eq:sr_model_taylor} \frac{dX}{dt} = ηt - β_{\text{SR}} \frac{X}{κ} + β_{\text{SR}}\frac{X²}{κ²} + \sqrt{2ε}ξ. \end{equation}

This is eq. \eqref{eq:generic_langevin} with

\begin{align} A &= η & B &= 0 & C &= \frac{β_{\text{SR}}}{κ} \\ G &= \frac{β_{\text{SR}}}{κ²} & D &= ε & & \end{align}

Later in sec. 10 we will discuss the validity of this Taylor expansion in more detail. In particular how it relates to the F&G model, the death criteria and the death threshold $X_c$in SR.

The F&G model in expanded form eq. \eqref{eq:fg_expanded} is,

\[ \frac{d z_0}{dt} = β_{\text{FG}}γt + β_{\text{FG}}Z_0 + J_0 - (ε_0 - β'Z_0) z_0 + gz²_0 + β'γt z_0 + \sqrt{2D_0} ξ \]

This lets us identify the equivalent coefficients compared to the generic Langevin equation \eqref{eq:generic_langevin} as,

\begin{align} A &= β_{\text{FG}}γ & B &= β_{\text{FG}}Z_0 + J_0 & C &= ε_0 - β'Z_0 \\ G &= g & D &= D_0 & & \end{align}

which only leaves us with one mismatched term. The $\beta \text{'}\gamma t{z}_0$term is a direct coupling between the time and position variables.

This leaves us with the initial conclusion of restricting us to the case of $\beta \text{'}=0$. With that restriction the F&G model reduces to,

\begin{equation} \label{eq:fg_beta_zero} \frac{d z_0}{dt} = β_{\text{FG}}γt + β_{\text{FG}}Z_0 + J_0 - ε_0 z_0 + gz²_0 + \sqrt{2D_0} ξ \end{equation}

Now compare the Taylor expanded version of the SR model, eq. \eqref{eq:sr_model_taylor} with the F&G variant without explicit time coupling, eq. \eqref{eq:fg_beta_zero}. We can then conclude the two models are equivalent if and only if,

\begin{align} η &= β_{\text{FG}}γ & Z_0 &= J_0 = 0 & β_{\text{SR}} &= ε_0 \\ \frac{β_{\text{SR}}}{κ²} &= g & ε &= D_{0} & & \end{align}

In other words, if we accept zero starting entropy $Z_0$and no constant stress $J_0$in the F&G model and restrict the parameter space of ${\epsilon }_0$to ${\epsilon }_0=g_{FG}{{\kappa }^2}_{SR}$(and thus reducing a degree of freedom as ${\epsilon }_0$and $g$are coupled via the $\kappa$parameter), the two models are identical for small $X$.

In order to relax the restriction that $Z_0=J_0=0$to obtain the same model as SR, we can make a minor extension to the SR model.

Namely, by performing a time shift \(t ↦ t + τ\), we gain an additional parameter, which can capture the constants. In general terms, the differential equation is,

\[ \frac{dX}{dt} = f(t, X(t)), \]

where $f$is the placeholder for the exact definition, which is solved by $X(t)$. If we define a new solution,

\[ Y(t) \coloneq X(t + τ), \]

we arrive at,

\[ \frac{dY}{dt} = \frac{d}{dt} X(t + τ) = f(t + τ, X(t + τ)) = f(t + τ, Y(t)). \]

That means, the time shifted equation with \(t ↦ t + τ\) is solved by $Y(t)$. Writing out this equation explicitly in our case,

\begin{equation} \label{eq:sr_time_shifted} \frac{dX}{dt} = η(t + τ) - β \frac{X}{κ} + β\frac{X²}{κ²} + \sqrt{2ε}ξ. \end{equation}

At this point the term $\eta \tau =B$corresponds to the constant offset. This time shifted version of the SR model can thus be mapped to the F&G model via either,

• $\eta \tau =Z_0,J_0=0$,
• $\eta \tau =Z_0+J_0$
• or $\eta \tau =J_0,Z_0=0$

depending on preference or purpose.

At this point we can map the F&G model to the SR model in an almost general fashion (with the only limitation still being $\beta \text{'}=0$).

If we map the coefficients of eq. \eqref{eq:generic_langevin} from the SR model with constant offset and the F&G model and solve them for the SR parameters, we arrive at the following mapping:

\begin{align} \label{eq:mapping_fg_sr} η &= β_{\text{FG}}γ & β_{\text{SR}} &= \frac{ε_0²}{g} \\ τ &= \frac{1}{γ}\left( Z_0 + \frac{J_0}{β_{\text{FG}}}\right) & κ &= \frac{ε_0}{g} \\ ε &= D_0 & & \end{align}

This allows to map any specific instantiation of the F&G model to an equivalent SR model.

Note that the $\tau$parameter, is independent of $a$. This makes a lot of sense, as the new time parameter $\tau$is just an offset of "starting damage" / "initial entropy". It does not depend on how damage accumulates over time or generally the system evolves.

We have now seen that with some minor modifications to the SR model, the two are almost identical. The one mismatch is the existence of the $\beta \text{'}$parameter in the F&G model, which couples its $z_0$variable to its $t$variable. We may wish to recover the lost degree of freedom from setting $\beta \text{'}=0$in the F&G model.

By taking $\beta$to be $\beta (t)$in the SR model in addition to the previous modifications, we can then also recover the coupling of time and position. In the simplest form we may take,

\[ β(t) = β_0 + β_1 t \]

We now plug this into our time shifted variant of the SR model, eq. \eqref{eq:sr_time_shifted}.

\begin{align} \frac{dX}{dt} &= η(t + τ) - β(t) \frac{X}{κ} + β(t)\frac{X²}{κ²} + \sqrt{2ε}ξ \\ &= η(t + τ) - β_0 \frac{X}{κ} + β_0 \frac{X²}{κ²} - β_1 t \frac{X}{κ} + β_1 t \frac{X²}{κ²} + \sqrt{2ε}ξ \end{align}

If we map ${\beta }_0$to the normal $\beta$we had before we now have two additional terms, $-{\beta }_{1}t\frac{X}{\kappa }+{\beta }_{1}t\frac{X^2}{{\kappa }^2}$. The former is the equivalent of the $\beta \text{'}{z}_0\gamma t$term in the F&G model. We are justified in ignoring the latter term, as the product $t{X}^2$is of a higher order than the terms we decided to include after our Taylor expansion, which we cut off at $X^3$. This means our final model is,

\begin{equation} \frac{dX}{dt} = η(t + τ) - β_0 \frac{X}{κ} + β_0 \frac{X²}{κ²} - β_1 t \frac{X}{κ} + \sqrt{2ε}ξ \end{equation}

If we include a time position coupling into the general Langevin equation \eqref{eq:generic_langevin} from above, we arrive at this equation with added parameter $T$:

\begin{equation} \label{eq:generic_langevin_with_time} \frac{d x}{d t} = A t + T t x + B - C x + G x² + \sqrt{2 D} ξ(t). \end{equation}

We can use this to map both the F&G model and SR model to these now six parameters.

It leads to the following expressions, which while appearing slightly more complicated are a logical extension of the mapping in sec. Mapping F&G to SR, eq. \eqref{eq:mapping_fg_sr}:

\begin{align} \label{eq:full_mapping_fg_sr} η &= β_{\text{FG}}γ & β_{\text{SR}} &= \frac{(ε_0 - β'Z_0)²}{g} \\ τ &= \frac{1}{γ}\left( Z_0 + \frac{J_0}{β_{\text{FG}}}\right) & κ &= \frac{(ε_0 - β' Z_0)}{g} \\ ε &= D_0 & β_1 &= -\frac{ε_0 - β' Z_0}{g} β' γ \end{align}

Essentially, we need to replace ${\epsilon }_0$by ${\epsilon }_0-\beta \text{'}{Z}_0$and we have an additional equation for the new parameter ${\beta }_1$.

This is our final parameter mapping. This is an almost fully general mapping between the SR and F&G models, including all parameters of the F&G model. This mapping maps all eight parameters of the F&G model to the now six parameters of the SR model.

Having discussed the mapping of our modified SR model to the F&G model, let's now come back to one of the biggest modifications / assumptions we made: the Taylor expansion of the $-\frac{X}{X+\kappa }$ term in the SR model. As mentioned previously, this Taylor expansion is valid for small values of $X$, because we expanded around \(X = 0\). Small here is relative to the magnitude of $\kappa$, which tends to be at $\kappa =0.5$. However, the SR model defines death as the damage crossing the critical damage threshold $X_c\approx 17.5$. Those kind of damage values are well outside the validity range of the Taylor expansion.

Fig. 1 compares the relative damage removal rate of the actual term $-\frac{X}{X+\kappa }$with the first three orders of the Taylor expansion, linear $-\frac{X}{\kappa }$, quadratic $-\frac{X}{\kappa }+\frac{X^2}{{\kappa }^2}$and additionally cubic $-\frac{X}{\kappa }+\frac{X^2}{{\kappa }^2}-\frac{X^3}{{\kappa }^3}$. We can see that for values up to around $\kappa \lesssim 0.2$the three expansions are roughly compatible. However, beyond that they start to diverge rapidly with the linear and cubic terms exploding, indicating the damage removal keeping up with the damage, while the quadratic term actually goes back down to zero (and technically towards $-\infty$).

This limited validity range directly explains several apparent differences between the two models and highlights their complementary strengths.

Crucially, the SR model yields sensible results that reproduce biological systems up to large damage values. In contrast, a model like F&G or our modified SR model does not behave sensibly at large damage values. SR thus has more robust global behavior.

While this may appear to point to an enormous problem in our mapping between the models, the (in my opinion) correct interpretation is a lot more interesting. The fact that the Taylor expanded version is not sensible for large values of $X$implies the F&G model must use a different definition of death out of necessity. So while the two models can look very similar at the level of the differential equation, their actual application to an 'agent simulation' is inherently different.

Effectively, the SR and F&G models are similar models, with some different choices driven by their different definitions. Viewing the F&G model as a local approximation of the SR model explains why the F&G model cannot have a global damage threshold as the definition for death. At the same time, the F&G model in practice contains more independent parameters and includes constant offsets.

The functional definition of SR's differential equation contains F&G as a local, small-$X$approximation. The additional parameters that F&G exposes can only be modeled after mild modification (time shift, time dependent $\beta (t)$).