Migration of activator peaks in the transversal direction.

<p>(<b>a</b>) snapshot of YS domain (the YS domain is growing over time as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0102718#pone-0102718-g007" target="_blank">figure 7</a>). Morphogen concentration is denoted as z-axis height. Substrate has relatively low values inside the growing rectangle, and relatively high outside the growing rectangle. Y equals to 1.0 inside the rectangle and equal to 0.0 outside that rectangle. The initial rectangular is 5 space steps wide by 10 space steps long. The length of the rectangular increases one space step every 10,000 time steps. Space step dx = 0.3, time step dt = 0.4dx². (<b>b, c</b>) profile of S along the dotted line as shown in panel a. The high/low value of S profile is 1.0/0.6 and 1.0/0.4 in panel b and c. Activator peaks migrate out of the YS domain in a left-right order and a symmetrical manner under condition b and c respectively.</p

Widening of the Y-stalk is required for tip splitting.

<p>(<b>a</b>) The first activator peak emerges at the growing tip. (<b>b</b>) forward migration of the leading activator peak produces elongation of the Y-stalk. (<b>c</b>) the leading activator splits into two daughter peaks as the Y-stalk becomes wider, shown by the arrow. (<b>d</b>) each of the daughter activator peaks, from the 1^st generation splitting, becomes the leading activator peak of the newly formed stalks. (<b>e</b>) 2^nd generation of tip splitting occurs when the daughter stalks get wide enough, shown by the arrow. (<b>f</b>) each of the granddaughter activator peaks becomes the leading activator peaks of the newly formed stalks, and migrates forward. Parameters:  = 0.002,  = 0.16,  = 0.04,  = 0.03,  = 0.0001,  = 0.02,  = 0.02,  = 1.0,  = 0.008,  = 0.1,  = 10,  = 0.02,  = 0.26,  = 0.06.</p

A/H dynamics in tip splitting.

<p>(<b>a</b>) A/H dynamics as a function of S, Y. When (S, Y) pairs fall into the crescent moon region, the A/H subsystem has a classic Turing instability (by a linear Turing-instability criterion, see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0102718#pone.0102718-Murray1" target="_blank">[15]</a> page 87). When the (S, Y) pairs are located below the moon region, the temporal behavior of the A/H subsystem is oscillatory. For other (S, Y) pairs, the A/H subsystem has a stable temporal response. The dotted line shows a typical trajectory for cell differentiation. The cell (S, Y) state goes from the bottom right, ‘walks across’ the crescent moon, and reaches the top left. (<b>b</b>) Sites of Turing-ready cells formed a strip at the growing tip. When the black strip grew wide enough, it splits into two. Parameters:  = 2.0,  = 0.04, dx = 0.01, dt = 0.4dx², time steps between figures is 5000dt. (<b>c</b>) Dispersion relation of k1, k2, and k3 corresponds to the chosen (S, Y) pairs in the crescent moon region numbed 1, 2, and 3, respectively.</p

Growing YS domain produces activator peak insertion.

<p>When we command the rectangular YS domain to extend over time (by setting the length of the rectangular to be a function of time. The initial length of the rectangular area is 10 space steps. The length increases by 1 space step every 10,000 time steps; the width of the rectangular is held constant at 5 space steps, space step dx = 0.3, and time step dt = 0.4 dx²), A/H dynamics forms the activator peak insertion. The left column is the change of activator spatial pattern over time, and the right column is the corresponding spatial pattern of YS domain. Time increases from top to bottom. The first activator peak appears at the open end the YS domain (marked by the asterisk). More activator peaks will be induced and emerge right behind the leading activator peak when the growth creates enough space (marked by double-arrows).</p

Mechanisms of Side Branching and Tip Splitting in a Model of Branching Morphogenesis

<div><p>Recent experimental work in lung morphogenesis has described an elegant pattern of branching phenomena. Two primary forms of branching have been identified: side branching and tip splitting. In our previous study of lung branching morphogenesis, we used a 4 variable partial differential equation (PDE), due to Meinhardt, as our mathematical model to describe the reaction and diffusion of morphogens creating those branched patterns. By altering key parameters in the model, we were able to reproduce all the branching styles and the switch between branching modes. Here, we attempt to explain the branching phenomena described above, as growing out of two fundamental instabilities, one in the longitudinal (growth) direction and the other in the transverse direction. We begin by decoupling the original branching process into two semi-independent sub-processes, 1) a classic activator/inhibitor system along the growing stalk, and 2) the spatial growth of the stalk. We then reduced the full branching model into an activator/inhibitor model that embeds growth of the stalk as a controllable parameter, to explore the mechanisms that determine different branching patterns. We found that, in this model, 1) side branching results from a pattern-formation instability of the activator/inhibitor subsystem in the longitudinal direction. This instability is far from equilibrium, requiring a large inhomogeneity in the initial conditions. It successively creates periodic activator peaks along the growing stalk, each of which later on migrates out and forms a side branch; 2) tip splitting is due to a Turing-style instability along the transversal direction, that creates the spatial splitting of the activator peak into 2 simultaneously-formed peaks at the growing tip, the occurrence of which requires the widening of the growing stalk. Tip splitting is abolished when transversal stalk widening is prevented; 3) when both instabilities are satisfied, tip bifurcation occurs together with side branching.</p></div

Different branching patterns produced by altering key parameters.

<p>(<b>a, b, c</b>) as one of the key parameters gradually increases, the branching mode produced by the full model changes from (a) side branching with left-right alternating order to (b) side branching with symmetry, and then to (c) tip splitting.  = 0.0001 and  = 0.025/0.08/1.0 from left to right respectively. (<b>d, e</b>) as another key parameter increases, the spatial distance between side branches increased.  = 0.025, goes from 0.000025 to 0.0002. (<b>f</b>) mixed pattern formation of tip splitting and side branching, when  = 0.85,  = 0.00003, and and  = 0.004. Parameters:  = 0.002,  = 0.16,  = 0.04,  = 0.03,  = 0.02,  = 0.02,  = 0.008,  = 0.1,  = 10,  = 0.02,  = 0.26,  = 0.06.</p

Simulation with YS domain having two open ends.

<p>(<b>a</b>) initial condition of Y and S: we place the rectangular YS domain in the center, with high concentrations (Y = 1.0 and S = 0.5) inside the rectangle (5 space steps wide ×80 space steps long) and low concentrations (Y = 0.0 and S = 0.0) outside that rectangle. (<b>b, c</b>) two activator peaks emerge simultaneously at the two open ends of the YS domain, marked by the asterisks. (<b>d, e, f, g</b>) these two activator peaks induce more activator peaks to form along the YS domain, marked by double-arrows, in a wave-like manner, until the YS domain is filled up. Parameters:  = 0.002,  = 0.16,  = 0.04,  = 0.03,  = 0.0001,  = 0.02,  = 0.26. Space step dx = 0.3, time step dt = 0.4dx².</p

Increased growth speed, increased periodicity of the activator peaks.

<p>In the simulation of A/H dynamics with growing YS domain, when we increased the growth speed by a factor of 5 and 25, more activator peaks evolve along the Y-stalk. Parameters:  = 0.002,  = 0.16,  = 0.04,  = 0.03,  = 0.00005,  = 0.02,  = 0.26. Space step dx = 0.3, time step dt = 0.4dx². The initial shape of the rectangular is 5 space steps wide by 10 space steps long. The speed with which the rectangular extends differs. Control: every 10,000 time steps extend one space grid; 5X: every 2000 time steps extend one space grid; 25X: every 400 time steps extend one space grid.</p

Increased , increased spacing between activator peaks.

<p>In the simulation of A/H dynamics, with Y and S having high concentrations inside the rectangle (5 space steps wide ×80 space steps long), and low concentrations outside (details see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0102718#pone-0102718-g004" target="_blank">figure 4a</a>). When increases, the spacing between activator peaks increased. Parameters:  = 0.002,  = 0.16,  = 0.04,  = 0.03,  = 0.02,  = 0.26,  = 0.00005(a), 0.0002(b), 0.00035(c), and 0.0004(d). Space step dx = 0.3, time step dt = 0.4dx².</p

Spatial pattern of each variable in side branching.

<p>The spatial pattern of activator A (<i>top left</i>) and inhibitor H (<i>top right</i>) overlap, while the Y-stalk (<i>bottom left</i>) and substrate S (<i>bottom right</i>) are spatially complementary. (morphogen concentrations are denoted as z-axis height).</p