Programming cell growth into different cluster shapes using diffusible signals

Abstract

Advances in genetic engineering technologies have allowed the construction of artificial genetic circuits, which have been used to generate spatial patterns of differential gene expression. However, the question of how cells can be programmed, and how complex the rules need to be, to achieve a desired tissue morphology has received less attention. Here, we address these questions by developing a mathematical model to study how cells can collectively grow into clusters with different structural morphologies by secreting diffusible signals that can influence cellular growth rates. We formulate how growth regulators can be used to control the formation of cellular protrusions and how the range of achievable structures scales with the number of distinct signals. We show that a single growth inhibitor is insufficient for the formation of multiple protrusions but may be achieved with multiple growth inhibitors, and that other types of signals can regulate the shape of protrusion tips. These examples illustrate how our approach could potentially be used to guide the design of regulatory circuits for achieving a desired target structure.

Fig 1. Schematic of the model.

(a) We consider a cluster of cells, each having the potential to secrete q different diffusible chemicals. The secretion rate ${\mu }_i\left(\overrightarrow{c}\right)$ of each of these chemicals i = 1, …, q, and the growth rate $g\left(\overrightarrow{c}\right)$ of each cell depend on the local external chemical environment $\overrightarrow{c}$ of the cell. Given the spatial profile of growth rate inside the cluster, we can then solve for the velocity of cell flow $\overrightarrow{u}$, which specifies how the shape of the tissue changes over time (blue arrows). (b) We initialize the system with a 2-dimensional parabolic cluster with initial length x₀ and initial width y₀.

Fig 2. Dynamics of cluster growth without growth regulation.

Cluster is initialized as (a) a parabolic cluster and (b) an elliptical cluster. As all cells grow at the same rate, an initial elongated cluster (black lines) becomes increasingly circular over time (black to pink to green).

Fig 3. Dynamics of a parabolic 2-D cluster with cells secreting a single growth inhibitor.

(a) We treat X as a growth inhibitor (as indicated by the elliptical arrowhead), such that cells can only grow if the concentration of X is below a threshold i.e. ${\tilde{c}}_X<1$. (b) Only cells at the tip of the tissue can grow, with the initial size of the growth zone (black region) decreasing with increasing effective secretion rate of inhibitor ${\tilde{\mu }}_X$. As cluster grows, a rod-like extension emerges. The different colored regions represent the growth zones at different times. (c) Rescaled inhibitor concentration ${\tilde{c}}_{X,min}$ at the tip initially increases but reaches a steady-state level where it stays approximately constant. (d) Area of growth zone eventually stays approximately constant as the rod-like extension grows. The steady-state area decreases with ${\tilde{\mu }}_X$. [Other parameters: initial tissue length ${\tilde{x}}_0=1.5$, initial tissue width ${\tilde{y}}_0=1$].

Fig 4. Dynamics of cluster when cells secrete two growth inhibitors.

(a) We consider here the case where both X and Y inhibits growth. In addition, X also inhibits the production of Y, such that its secretion rate ${\tilde{\mu }}_Y\left({\tilde{c}}_X\right)$ is a threshold function with Y only produced near the tip of the cluster where ${\tilde{c}}_X<{\tilde{K}}_s$. (b-d) The initial shape of the growth zone (in red) depends on the values of secretion threshold ${\tilde{K}}_s$ and ${\tilde{\mu }}_Y$ through concentration profile of Y (S4 Fig). The blue regions consist of arrows perpendicular to the tissue boundary, with the length of the arrows proportional to the boundary velocity at that point. (b) When ${\tilde{K}}_s=0.9$, the region of the growth zone closer to the tissue tip stops growing first as ${\tilde{\mu }}_{Y0}$ is increased (left: ${\tilde{\mu }}_{Y0}=150$, right: ${\tilde{\mu }}_{Y0}=200$). (c) When ${\tilde{K}}_s=1$, growth zone depletes from its center as ${\tilde{\mu }}_{Y0}$ increases (left: ${\tilde{\mu }}_{Y0}=56$, middle: ${\tilde{\mu }}_{Y0}=60$, right: ${\tilde{\mu }}_{Y0}=65$). (d) When the ratio of secretion threshold to growth threshold ${\tilde{K}}_s=1.2$, the growth zone shrinks from the left boundary as ${\tilde{\mu }}_{Y0}$ increases (left: ${\tilde{\mu }}_{Y0}=20$, right: ${\tilde{\mu }}_{Y0}=22$). The growth zone therefore remains attached to the tip of the tissue. (e-g) Cluster dynamics for the different values of ${\tilde{K}}_s$, with the different colored regions representing the growth zones at different time points (colors randomly generated). (e) When ${\tilde{K}}_s=0.9$, the cluster grows a single rod-like protrusion even though there are initially multiple growth regions. (f) When ${\tilde{K}}_s=1$, cluster can grow multiple protrusions (S5 Fig). (g) When ${\tilde{K}}_s=1.2$, there is only a single growth zone and hence a single protrusion. [Other dimensionless parameters: ${\tilde{\mu }}_X=8$, γ_r = 1.].

Fig 5. Dynamics of cluster when cells secrete a growth threshold regulator in addition to a growth inhibitor.

(a) Y reduces the growth threshold of X. Here, we have chosen the rescaled growth threshold ${\tilde{K}}_g=\frac{K_g}{K_{g0}}$ to decrease linearly with the rescaled concentration ${\tilde{c}}_Y=a{c}_Y$. (Eq 9) (b) The cluster grows a rod-like structure when all cells secrete Y at a constant effective rate ${\tilde{\mu }}_Y=\frac{a{\mu }_Y}{{\gamma }_1}\frac{D_1}{D_2}$. (c) We consider the scenario where X inhibits the production of Y such that only a region at the tip of the tissue can secrete Y. (d) For the scenario described in (c), it is possible for the protrusion to become narrower over time if the secretion threshold is less than the growth threshold ${\tilde{K}}_s=\frac{K_s}{K_{g0}}<1$ (top: ${\tilde{\mu }}_Y=6$, bottom: ${\tilde{\mu }}_Y=15$). (e) We also allow the maximum secretion rate of Y to decrease linearly with c_X. (Eq 10). (f) When ${\tilde{K}}_s=0.9$, a stronger regulation of μ_Y,max (more negative $\tilde{b}$) produces sharper cone-like structures (top: ${\tilde{\mu }}_Y=15$, middle: ${\tilde{\mu }}_Y=26.5$, bottom: ${\tilde{\mu }}_Y=48$). [Other dimensionless parameters: ${\tilde{\mu }}_X=8$, γ_r = 0.2.].