Analytic Study of Three-Dimensional Single Cell Migration with and without Proteolytic Enzymes

Abstract

Cell motility is a fundamental physiological process that regulates cellular fate in healthy and diseased systems. Cells cultured in 3D environments often exhibit biphasic dependence of migration speed with cell adhesion. Much is not understood about this very common behavior. A phenomenological model for 3D single-cell migration that exhibits biphasic behavior and highlights the important role of steric hindrance is developed and studied analytically. Changes in the biphasic behavior in the presence of proteolytic enzymes are investigated. Our methods produce a framework to determine analytic formulae for the mean cell speed, allowing general statements in terms of parameters to be explored, which will be useful when interpreting future experimental results. Our formula for mean cell speed as a function of ligand concentration generalizes and extends previous computational models that have shown good agreement with in vitro experiments.

Appendices

Appendix 1

We show here how the computational models of previous studies12, 37 can be given a precise mathematical formulation that enables the mean cell speed as a function of system parameters such as ligand density to be computed in terms of simple known functions, circumventing the need for simulation. The analytical formulae obtained require no restriction on the value of the force ratio parameter $\epsilon$ and the nature of the approximation in which the protrusion force is neglected is clearly revealed, demonstrating its innocence in the usual biologically relevant parameter regime.

In three-dimensions, we introduce Cartesian basis vectors $\widehat{\mathbf{i}}, \widehat{\mathbf{j}}$ and $\widehat{\mathbf{k}}$, with $\widehat{\mathbf{k}}$ in the direction of $\mathbf{F}_{\rm trac}$, so $\mathbf{F}_{ \rm trac}=F_{\rm trac}\widehat{\mathbf{k}}$ and with 0 ≤ θ ≤ π and 0 ≤ ϕ ≤ 2π,

$$ {\mathbf{F}}_{\rm prot}=F_{\rm prot}\sin\theta(\cos\phi \widehat{{\mathbf{i}}}+\sin\phi \widehat{{\mathbf{j}}}) +F_{\rm prot}\cos\theta\widehat{{\mathbf{k}}}. $$

Since the integration element for isotropic random angles is $(4\pi)^{-1}\sin\theta d\theta d\phi$, the calculation of ${\mathbb{E}\{ |\mathbf{v} |\}}$ as a double integral is straightforward and the final answer is

$$ {\mathbb{E}}\{|{\mathbf{v}}|\}=\left\{ \begin{array}{ll} \frac{ F_{\rm trac}^2+3F_{\rm prot}^2}{3H(L)cF_{\rm prot}} & \hbox{if}\ F_{\rm prot} > F_{\rm trac}, \\ \frac{F_{\rm prot}^2+3F_{\rm trac}^2}{3H(L)cF_{\rm trac}} &\hbox {if}\ F_{\rm trac}>F_{\rm prot}. \end{array}\right. $$

(20)

It may be noted that if we desired the probability density functions for $|\mathbf{v}|$ rather than just their expected values, the problem is equivalent to a two-step Rayleigh random flight with unequal step lengths (see Hughes15 for details).

Any modeling of MMP activity that alters the values of one or more of F _trac, F _prot, or H(L) can easily be accommodated in Eq. (20). As it will almost always be the case that F _trac > F _prot, we will usually have

$$ {\mathbb{E}}\{|{\mathbf{v}}|\}= \frac{ F_{\rm prot}^2+3F_{\rm trac}^2}{3H(L)cF_{\rm trac}} =\frac{F_{\rm trac}}{H(L)c}\left[1+\frac{1}{3}\left(\frac{F_{\rm prot}}{F_{\rm trac}}\right)^2\right] $$

(21)

and often (as in case studies in the “Results” section), we have F _prot/F _trac≪ 1, leading to a further simplification. However, in our discussion of a time-evolving simulation approach12, 37 in which the ligand density is very greatly reduced towards the end of the simulation time interval, we have used Eq. (20).

Appendix 2

We sketch the analysis of the location of the velocity maximum in Eq. (15) for the prescription (16) of the effects of MMPs. We have

$$ u(\ell)=\ell(1-\ell)(1+\alpha\delta\ell)\exp[\alpha(\gamma-\beta)\ell], $$

so u(0) = u(1) = 0 and u(ℓ) > 0 for 0 < ℓ < 1, ensuring that the continuous function u(ℓ) attains at least one local maximum inside the interval. Taking the natural logarithm and differentiating twice, we find that

$$ \frac{u'(\ell)}{u(\ell)}=\frac{1}{\ell}+\frac{1}{\ell-1}+ \frac{\alpha\delta}{1+\alpha\delta\ell}+\alpha(\gamma-\beta) $$

and

$$ \frac{u''(\ell)}{u(\ell)}-\frac{u'(\ell)^2}{u(\ell)^2}=-\frac{1}{\ell^2}- \frac{1}{(\ell-1)^2}-\frac{(\alpha\delta)^2}{(1+\alpha\delta\ell)^2}. $$

We see that at relevant stationary points [i.e., 0 < ℓ^* < 1 and u′(ℓ^*) = 0] we have u′′(ℓ^*) < 0, so there is always exactly one such stationary point, which is a both a local maximum and the global maximum of u(ℓ) for 0 ≤ ℓ ≤ 1. Its location is the unique solution in 0 < ℓ < 1 of the algebraic equation

$$ \frac{1}{\ell}+\frac{1}{\ell-1}+ \frac{\alpha\delta}{1+\alpha\delta\ell}+\alpha(\gamma-\beta)=0. $$

(22)

If αδ = 0 or if α(β − γ) = 0 the equation is quadratic and ℓ^* can be exhibited in simple form. In all other cases ℓ is given by solving a cubic equation, though we refrain from writing out the solution here. The limiting behavior of the solution in any of the limits $\delta\to\infty$, α(β − γ)→ 0 and $\alpha(\beta-\gamma)\to-\infty$ can easily be extracted from Eq. (22). Details will be found in Table 2.

Although for biological relevance one needs all of α, β, γ and δ to be non-negative, the conclusions we have just drawn concerning the maximum are true on the weaker assumption that αδ >  − 1 (which is needed to ensure that receptor density remains positive), with no restriction on the signs or magnitudes of α, β, γ or δ. Setting α = 0 completely removes the effects of MMPs and leaves the speed maximal at ℓ = 1/2, so in the analysis summarized in Table 2 we assume that α > 0 and in some cases for brevity we write ξ = α(β − γ).