Modeling and analysis of melanoblast motion

Abstract

Melanoblast migration is important for embryogenesis and is a key feature of melanoma metastasis. Many studies have characterized melanoblast movement, focusing on statistical properties and have highlighted basic mechanisms of melanoblast motility. We took a slightly different and complementary approach: we previously developed a mathematical model of melanoblast motion that enables the testing of biological assumptions about the displacement of melanoblasts and we created tests to analyze the geometric features of cell trajectories and the specific issue of trajectory interactions. Within this model, we performed simulations and compared the results with experimental data using geometric tests. In this paper, we developed the associated mathematical model and the main focus is to study the crossings between trajectories with new theoretical results about the variation of number of intersection points with respect to the crossing times. Using these results it is possible to study the random nature of displacements and the interactions between trajectories. This analysis has raised new questions, leading to the generation of strong arguments in favor of a trail left behind each moving melanoblast.

Appendix

We need the following theorem which is a particular case of formula p. 561 of Morton (1966) (see also Solomon 1978, page 16 chap. 1) and is an extension of the famous Buffon needle problem :

Theorem 2

Let two rectifiable curves ${C}_1$ and $C_2$ with length $L_i, i=1,2$. Let $\varGamma _1$ and $\varGamma _2$ two curves obtained by displacements $D_{x_i,y_i,\theta _i}$ of ${C}_i, i=1,2$ where $x_i,y_i,\theta _i$ are uniformly distributed variable in $\mathbb {R}/\mathbb {Z}$. The expected value ${\mathcal {E}}({\mathcal {C}})$ of the number of intersection points between two curves of $\varGamma _1$ and $\varGamma _2$ is

$$\begin{aligned} {\mathcal {E}}({\mathcal {C}}) = \frac{2}{\pi } L_1L_2. \end{aligned}$$

(2)

Note that the result also applies to any parts of curves $C_1$ and $C_2$.

1.1 4.3.1 Proof of Theorem 1

We start by proving Theorem 1 for $p=2$. For $t \in [\sigma , 1 ]$ let us denote ${\mathcal {E}}_{\sigma }({\mathcal {C}},t)$ the expected value of the number of intersection points between two trajectories of $\varGamma _1$ and $\varGamma _2$ whose parameters $t_1$ and $t_2$ are such that $|t_1-t_2| \le \sigma$ and $t_1,t_2 \in [0,t]$. Recall that a trajectory is a piecewise linear function on a partition of [0, 1] defined by the times $t^k, k=1,...,N$ and that a trajectory is traveled at constant speed. Thus the length of a part of a trajectory ${\mathcal {C}}_i$ defined by an interval $[t,t+\varDelta t]$ is $\varDelta t L_i$. Let us compare ${\mathcal {E}}_{\sigma }({\mathcal {C}},t)$ and ${\mathcal {E}}_{\sigma }({\mathcal {C}},t+dt)$ for $t, t+dt \in [t^k,t^{k+1}]$. The set of intersection points between two trajectories, defined by times $t_1, t_2 \in [0,t+dt]$ and $|t_1-t_2| \le \sigma$, is the disjoint union of four subsets:

• 1. $t_1, t_2\in [0,t]$ and $|t_1-t_2| \le \sigma$, hence the expected value of this subset of intersection points is ${\mathcal {E}}_{\sigma }({\mathcal {C}},t)$

• 2. $t_1\in [t-\sigma ,t]$ and $t \le t_2\le t+dt$ and $|t_1-t_2| \le \sigma$, this subset of intersection points is contained in the subset such that $t_1\in [t -\sigma ,t]$ and $t \le t_2\le t+dt$ and contains the subset such that $t_1\in [t-\sigma +dt,t]$ and $t \le t_2\le t+dt$. The first subset is the set of intersection points of a curve of length $\sigma L_1$ and a curve of length $dt L_2$. The second subset is the set of intersection points of a curve of length $(\sigma -dt)L_1$ and a curve of length $dt L_2$. Hence, using Theorem (2), the expected value of intersection points in this set is bounded below by $\frac{2}{\pi }(\sigma -dt) dt L_1L_2$ and upper by $\frac{2}{\pi }\sigma dt L_1L_2$ and thus is equivalent to $\frac{2}{\pi }\sigma dt L_1L_2$ as dt goes to 0.

• 3. $t_2\in [t - \sigma ,t]$ and $t \le t_1\le t+dt$ and $|t_1-t_2| \le \sigma$, this case is the same as the previous one by transposing the indexes.

• 4. $t \le t_1, t_2\le t+dt$. We can assume $|t_1-t_2| \le \sigma$ if dt is small enough and thus the expected value of intersection points is in this case $dt^2 L_1 l_2$. This last term can be neglected with respect to the previous ones when dt goes to 0.

Therefore as dt goes to 0 the limit of the quotient $\frac{{\mathcal {E}}_{\sigma }({\mathcal {C}},t+dt) -{\mathcal {E}}_{\sigma }({\mathcal {C}},t) }{dt}$ is given by summing cases 2 and 3 and thus is $\frac{4}{\pi }\sigma L_1L_2$. We have proved

$$\begin{aligned} \frac{d}{dt} {\mathcal {E}}_{\sigma }({\mathcal {C}},t) =\sigma \frac{4}{\pi }L_1L_2. \end{aligned}$$

After integration between $t=\sigma$ and $t=1$, we get

$$\begin{aligned} {\mathcal {E}}_{\sigma }({\mathcal {C}},1) = {\mathcal {E}}_{\sigma }({\mathcal {C}},\sigma ) + \sigma (1-\sigma )\frac{4}{\pi } L_1L_2. \end{aligned}$$

For $t = \sigma$, ${\mathcal {E}}_{\sigma }({\mathcal {C}},\sigma )$ is the expected value of number of intersection points between the two initial parts of trajectories $\varGamma _1$ and $\varGamma _2$ defined by $0\le t\le \sigma$ and thus it is the expected value of number of intersection points between two curves of length $\sigma L_1$ and $\sigma L_2$. Hence by Theorem (2)

$$\begin{aligned} {\mathcal {E}}_{\sigma }({\mathcal {C}},\sigma ) = \sigma ^2\frac{2}{\pi } L_1L_2, \end{aligned}$$

Finally, since ${\mathcal {E}}_{\sigma }({\mathcal {C}}) = {\mathcal {E}}_{\sigma }({\mathcal {C}},1)$,

$$\begin{aligned} {\mathcal {E}}_{\sigma }({\mathcal {C}}) = (2\sigma - \sigma ^2) \frac{2}{\pi } L_1L_2. \end{aligned}$$

Now, let us consider a set ${\mathcal {C}}$ of p trajectories with length $L_i, i=1,\cdots ,p$. The set of intersection points between different randomly displaced trajectories from this set is a disjoint union of the number of intersection points between two trajectories and thus we get formula 1, taking into account that each pair of curves is counted twice in the right hand side sum. $\square$

1.2 4.3.2 Proof of Corollary 1

Let a constant speed random trajectory defined by a function f(t). By Definition (1) each segment $f([t^k,t^{k+1}])$ has a fixed length l and is defined by a random angle $2\pi \theta ^k$ with $\theta ^k$ uniformly distributed over [0, 1]. By induction it is clear that the segment $f([t^k,t^{k+1}])$ is the image of a fixed segment of length l and angle 0 by a uniformly random displacement $D_{x^k,y^k,\theta ^k}$ with $x^k+iy^k=f(t^k)$: by Definition f(0) is uniformly distributed in $\varOmega$ and if $f(t^k)$ is uniformly distributed, then $f(t^{k+1})=f(t^k)+ l e^{2i\pi \theta ^k}$ is also uniformly distributed. Therefore the above proof of Theorem (1) is still valid because the Theorem (2) can still be applied to the comparison between ${\mathcal {E}}_{\sigma }(t)$ and ${\mathcal {E}}_{\sigma }(t+dt)$. $\square$