In this section we introduce the model assumptions and show in detail the development of the model and the derivation of the main result, given by the powerlaw distribution in the number of solute molecules trapped inside the liposomes.
Model assumptions
The lipidsolute interaction, discussed in the previous Section, is encoded in the capacity of the liposome to trap solute molecules, i.e., in the flux rate of molecules entering into the forming liposome through its open borders. Given the reasonable assumption that the probability of trapping is proportional to the area of the open surface of the liposome (permeability), the flux rate of solute turns out to be proportional to the closing rate r(τ ) of the liposome borders, being τ the time elapsed since the beginning of the liposome formation. The ferritinlipid interaction hypothesis implies that the closing rate and, consequently, the ferritin flux rate, is affected by the passage of ferritin itself.
It is worth noting that the assumption of a ferritinlipid interaction affecting the liposome closing rate is not incompatible with a possible anomalous diffusion of ferritin and/or lipid molecules inside water coupled with some clustering mechanism. However, this assumption needs a experimental validation and, consequently, the setup of new experimental work and will be the subject of future investigations.
We also note that, if the motion of solute particles in the aqueous solution was completely random, the net flux rate of solute on the liposome open borders would be zero. As a consequence, an asymmetric mechanism driving the flux rate must be assumed in order to get an average net flux of solute inside the liposome.
The asymmetric mechanism should involve a interaction ferritinferritin (clustering) and/or ferritinlipid that changes depending on the position of the ferritin (inside or outside the forming vesicle). This asymmetry must determine a probability, for a ferritin molecule, of moving from the external environment into the vesicle greater than the opposite direction. Between these two hypotheses, the ferritinlipid interaction might be more realistic. Indeed, soluble proteins such as ferritin do not clusterize spontaneously because of the composition of their surface which makes the interactions with water molecules thermodynamically more favourable than the proteinprotein interaction.
In summary, our model is grounded on two hypotheses:

The "jamming hypothesis" [2–4]: the interactions between the particles of solute and the incomplete vesicles interfere with (jam) the process of liposome formation; in particular the closing of liposomes is slowed.

The "semipermeability" hypothesis: the diffusion of solute particles from the solution towards the inside of the forming liposomes is faster than the diffusion in the opposite direction; in other words, there is a net flux of particles directed inside the forming vesicle.
These two hypotheses must be considered together with the renewal property, which is a reasonable assumption if each liposome does not interact with the other ones, that is:

(i)
there are so many lipids that there's no competition in the formation of lipid vesicles or liposomes;

(ii)
there are so many ferritin molecules and the liposomes are far enough from each other that there's no competition in the trapping of ferritin.
Then, the applicability of the renewal condition is exactly the same as in the Cox book [5] by substituting the failure events of some electronic devices, described there, with our closing events of some liposomes. In fact, under the assumptions (i) and (ii), the statistical ensemble of liposomes becomes a set of statistically independent realizations, so that both closing events and the associated closing times τ elapsed from the beginning of the experiment are mutually independent random variables and linear averaging can be applied to define mean quantities.
In the framework of renewal point processes, this model seems to be the minimal one explaining the emerging powerlaw behavior.
In fact, if we do not make this minimal assumption, the dynamics of lipids and ferritin would be independent and the flux of ferritin molecules across the liposome borders would be totally random, thus giving rise to a Poisson distribution of trapped ferritin molecules. Note that the renewal condition is associated with the emergence of an asymptotic selforganized structure (closed liposome + trapped ferritin molecules) following the occurrence of the closing event, i.e., the crucial event after which no flux of ferritin across the liposome surface is no longer possible. The renewal condition is then a natural hypothesis for the closing event, as the final state given by the asymptotic structure is well defined, but there's no way to know the exact time evolution that brought the system dynamics towards that particular asymptotic, stationary state.
In the next subsection we will show how, starting from these two very basic phenomenological hypotheses, it is possible to derive the observed powerlaw behavior.
Derivation of the model
Given the assumptions and hypotheses stated above, we will deal with the problem of finding the form for the event rate r(t) which is consistent with the experimentally observed powerlaw behavior for the distribution of ferritin molecules trapped inside the liposomes.
The jamming hypothesis can be formalized as follows:
{r}_{c}\left(t\right)=\frac{{r}_{0}}{\mathsf{\text{1}}+N\left(t\right)},
(9)
being N (t) the number of ferritin molecules trapped inside the liposome at time t. This is almost a natural choice, as we require a rate function that must slow down when N (t) increases, while the unit in the denominator avoid singularities at t = 0. r_{0} is a dimensional constant representing the closing rate of the liposomes in absence of ferritin. N (t) is not a deterministic function, monotonically increasing with time, rather it is a fluctuating random variable with an average tendency to increase. In other words, the semipermeability hypothesis imposes that the average number of ferritin molecules \u3008N\u3009(t) is a increasing function of time t. The rate r_{
c
}(t), given in Eq. (9), is also a random variable, as it depends on the random variable N (t). Essentially, r_{
c
}(t) is the closing rate associated with the stochastic dynamics of a single liposome interacting with the surroundings solute molecules (ferritin and lipids), while, under the renewal assumption, the average rate \u3008{r}_{c}\u3009(t) describes the average behavior of the total ensemble of independent systems. Each system is composed by a liposome approaching the closing event and by the surrounding ferritin molecules randomly passing through its surface.
In order to derive a formula for the average rate \u3008{r}_{c}\u3009(t), we consider the following expression:
N\left(t\right)=\u3008N\u3009\left(t\right)+\mathrm{\Delta}N\left(t\right),
(10)
where we introduced the zeromean fluctuation ΔN (t). Let us assume, without any loss of generality, that N (t) is a Poisson counting process. Then, the following well known result applies:
\u3008{\left(\mathrm{\Delta}N\right)}^{\mathsf{\text{2}}}\u3009\left(t\right)=\u3008N\u3009\left(t\right)\Rightarrow \mathrm{\Delta}N\left(t\right)~\sqrt{\u3008N\u3009\left(t\right)}.
(11)
Eq. (9) can be rewritten in the following way:
{r}_{c}\left(t\right)=\frac{{r}_{0}}{\mathsf{\text{1}}+\u3008N\u3009\left(t\right)+\mathrm{\Delta}N\left(t\right)}=\frac{{r}_{0}}{\mathsf{\text{1}}+\u3008N\u3009\left(t\right)}\frac{1}{\mathsf{\text{1}}+\mathrm{\Delta}N\left(t\right)/\u3008N\u3009\left(t\right)},
(12)
For time t large enough, the ratio \mathrm{\Delta}N\left(t\right)/\u3008N\u3009\left(t\right) becomes negligible due to Eq. (11), which involves \mathrm{\Delta}N\left(t\right)/\u3008N\u3009\left(t\right)~\mathsf{\text{1}}/\sqrt{\u3008N\u3009\left(t\right)}. Then, we have:
{r}_{c}\left(t\right)\simeq \frac{{r}_{0}}{\mathsf{\text{1}}+\u3008N\u3009\left(t\right)}\left(\mathsf{\text{1}}\frac{\mathrm{\Delta}N\left(t\right)}{\u3008N\u3009\left(t\right)}\right),
(13)
and, making the average of both sides, we finally get:
\u3008{r}_{c}\u3009\left(t\right)=\frac{{r}_{0}}{\mathsf{\text{1}}+\u3008N\u3009\left(t\right)},
(14)
with \u3008\mathrm{\Delta}N\u3009\left(t\right)=0 by definition. In summary, the average closing rate \u3008{r}_{c}\u3009 of a given liposome depends on \u3008N\left(t\right)\u3009, i.e., on the average number of solute particles present in the vesicle at time t. We note that, for the above derivation of Eq. (14) it is sufficient to assume a negligible fluctuation ΔN (t) for large times t, without resorting to the assumption of a Poisson process for N (t). On the contrary, the semipermeability hypothesis is a fundamental one, as it allows to derive a simple linear expression for N (t). This hypothesis simply states that, while liposomes are forming, the flux of solutes per unit of time from the environment towards the inside of the vesicles (λ_{
in
}) is greater than the flux in the opposite direction (λ_{
out
}). It is reasonable to assume that the relative slowdown of the outward flux can be due to some interactions between the solute molecules and the lipids composing the inner face of the vesicle. Thus, there is a net flux of solutes per unit of time (λ = λ_{
in
} − λ_{
out
}) directed towards the inner part of the forming vesicle. As a consequence, making an additional, but reasonable, linear assumption, we get:
< N (t) >= λt, and Eq. (14) becomes:
\u3008{r}_{c}\u3009\left(t\right)=\frac{{r}_{0}}{\mathsf{\text{1}}+\lambda t}.
(15)
This equation is formally identical to Eq. (7), so that the average Survival Probability Function of the liposome closing times is given by Eq. (8). The PDF describing the distribution of closure times is easily obtained by deriving Eq. (8):
\psi \left(t\right)=\frac{d\mathrm{\Psi}}{dt}=\frac{{r}_{0}}{{\left(\mathsf{\text{1}}+\lambda t\right)}^{\mathsf{\text{1}}+{r}_{0}/\lambda}}
(16)
Note that both Eqs. (8) and (16) become a purely inverse powerlaw distributions in the limit of large times: t >> 1/λ = T .
The distribution of ferritin molecules trapped inside liposomes: P (N ), is computed by applying the relationship defining the probability distribution of the function of a random variable, which is a random variable itself. In this case, the random function is given by N = N (τ ), being τ the closing times whose PDF is given by ψ(t). Then, approximating N with a continuous variable in the range of large N, we have:
P\left(N\right)dN=\psi \left(t\right)dt,
(17)
and, substituting Eq. (16), we get the following expression:
P\left(N\right)=\frac{\left(\mu \mathsf{\text{1}}\right)}{{\left(\mathsf{\text{1}}+N\right)}^{\mu}},
(18)
being:
\mu =\mathsf{\text{1}}+\frac{{r}_{0}}{\lambda}.
(19)
The derivation of the expression for P (N ) and the relationship given in Eq. (19), relating the exponent of P (N ) with the parameters of the closing rate function (r_{
c
})(t) represent the main result of this work. As a preliminary validation, we made a fit with data taken from the paper of Luisi et al. [2]. Fitting Eq. (18) with the data, we obtain the best agreement for µ = 2.3, as shown in Figure 2.