Multi-scale modeling of Snail-mediated response to hypoxia in tumor progression

TL;DR

A novel multi-scale mathematical model is proposed to analyze the pivotal role of Snail protein expression in the cellular responses to hypoxia and sheds light on the potential of the mathematical framework in advancing the understanding of the biological mechanisms driving tumor progression.

Abstract

Tumor cell migration within the microenvironment is a crucial aspect for cancer progression and, in this context, hypoxia has a significant role. An inadequate oxygen supply acts as an environmental stressor inducing migratory bias and phenotypic changes. In this paper, we propose a novel multi-scale mathematical model to analyze the pivotal role of Snail protein expression in the cellular responses to hypoxia. Starting from the description of single-cell dynamics driven by the Snail protein, we construct the corresponding kinetic transport equation that describes the evolution of the cell distribution. Subsequently, we employ proper scaling arguments to formally derive the equations for the statistical moments of the cell distribution, which govern the macroscopic tumor dynamics. Numerical simulations of the model are performed in various scenarios with biological relevance to provide insights into the role of the multiple tactic terms, the impact of Snail expression on cell proliferation, and the emergence of hypoxia-induced migration patterns. Moreover, quantitative comparison with experimental data shows the model's reliability in measuring the impact of Snail transcription on cell migratory potential. Through our findings, we shed light on the potential of our mathematical framework in advancing the understanding of the biological mechanisms driving tumor progression.

Figures (8)

• Figure 1: Experiment 1: initial conditions. Left: initial Gaussian distribution of the tumor cells $M^0$, centered in $\mathbf{x}_M = [9,9]\,\text{mm}$ in the domain $\Omega=[0,50]\times[0,50]\,\text{mm}^2$, together with the level plot for the fixed Gaussian distribution of oxygen $O_2({\bf x})$, centered in $\mathbf{x}_O = [45,45]\,\text{mm}$. Right: 1D profiles of tumor (continuous line) and oxygen (dashed line) distributions along the bisecting line (light gray line in the 2D plot) of the domain $\Omega$. $\bar{x}$ indicates the spatial position along this bisecting line.
• Figure 2: Experiment 1: spatial distribution of the environmental signals triggering cell motility. Graphical representation of the 1D section defined by the bisecting line of the domain $\Omega$ and its subdivision into four different areas, depending on the combination of high/low motility, random/directed motion, and low/high proliferation capability of the tumor cells. Profiles of the oxygen distribution (black continuous line) and tactic sensitivity $F(y^{*},O_2(\bar{x}))$ (blue dashed line) are shown, together with the oxygen gradient direction (red arrow). $\bar{x}$ indicates the spatial position along this bisecting line.
• Figure 3: Experiment 1: chemotactic- or anti-crowding-driven motion. Evolution of model \ref{['sim_M_ups']} in a chemotactic-driven ($\beta=0.98$, left column) or anti-crowding-driven ($\beta=0.8$, right column) scenario. Top row: evolution of the tumor mass in the domain $\Omega=[0,50]\times[0,50]\,\text{mm}^2$ at four different time steps, i.e., initial time $t=0$d (continuous light pink line), and progression at $t=6.25$d (dot pink line), $t=12.5$d (dashed dark pink line), and $t=20.8$d (dot-dashed purple line). Middle row: 1D profiles of the tumor mass evolution along the bisecting line (light gray line in the 2D plot) of the domain $\Omega$ at the same time steps used in the top row. $\bar{x}$ indicates the spatial position along this bisecting line. Bottom row: evolution in the $(t,y)$-space of the distribution $c_{y^*}(t,y)$, together with its mean $\eta(t)$, standard deviation $\sigma(t)$, and mode $\psi(t)$. References to the four selected time steps are repeated in each graph using consistent color and line styles. Parameter values are set as reported in Table \ref{['parameter_const']}.
• Figure 4: Experiment 2: impact of Snail expression on cell proliferation. Top row: 1D profiles representing the evolution along the bisecting line of $\Omega$ of the difference between the solution of the tumor equation in \ref{['sim_M_ups']} with the proliferation term given in \ref{['Macro_prol_1']} ($M_{\mathbb{P}_1}(t,\bar{x})$) and the solution of tumor equation in \ref{['sim_M_ups']} with the proliferation term given in \ref{['Macro_prol_2']} ($M_{\mathbb{P}_2}(t,\bar{x})$) at four different time steps, i.e., $t=5.2$d, $t=10.4$d, $t=15.6$d, and $t=20.8$d. $\bar{x}$ indicates the spatial position along this bisecting line. Bottom row: quantification of the percentage tumor mass increment (left plot) and the velocity of the center of mass (right plot) over time with the two choices of the proliferative operator. The continuous gray line refers to the choice \ref{['Macro_prol_1']}, while the dashed black line to \ref{['Macro_prol_2']}. Vertical lines in the bottom-row plots mark the selected times depicted in the first row. References to the four selected time steps are repeated in each graph using consistent color and line styles. Parameter values are set as reported in Table \ref{['parameter_const']}.
• Figure 5: Experiment 3: initial conditions and setting. 1D graphical illustration of the setting implemented for studying the impact of Snail expression and hypoxia on cancer cell migration. The domain $\Omega$ is divided into two chambers. Tumor cells are distributed in the upper chamber, i.e., in $\Omega_U=[0,25]\times[0,50]\,\text{mm}^2$, accordingly to \ref{['M_IC_chamber']}, while no cells are initially located in the bottom chamber, i.e., in $\Omega_B=[25,50]\times[0,50]\,\text{mm}^2$. Two different linear oxygen distributions (for normoxia and hypoxia scenarios) are represented as dashed black lines. The central membrane dividing the two chambers is shown in green.