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Dev Dyn
2006 Aug 01;2358:2144-59. doi: 10.1002/dvdy.20870.
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Mathematical model of morphogen electrophoresis through gap junctions.
Esser AT
,
Smith KC
,
Weaver JC
,
Levin M
.
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Gap junctional communication is important for embryonic morphogenesis. However, the factors regulating the spatial properties of small molecule signal flows through gap junctions remain poorly understood. Recent data on gap junctions, ion transporters, and serotonin during left-right patterning suggest a specific model: the net unidirectional transfer of small molecules through long-range gap junctional paths driven by an electrophoretic mechanism. However, this concept has only been discussed qualitatively, and it is not known whether such a mechanism can actually establish a gradient within physiological constraints. We review the existing functional data and develop a mathematical model of the flow of serotonin through the early Xenopus embryo under an electrophoretic force generated by ion pumps. Through computer simulation of this process using realistic parameters, we explored quantitatively the dynamics of morphogen movement through gap junctions, confirming the plausibility of the proposed electrophoretic mechanism, which generates a considerable gradient in the available time frame. The model made several testable predictions and revealed properties of robustness, cellular gradients of serotonin, and the dependence of the gradient on several developmental constants. This work quantitatively supports the plausibility of electrophoretic control of morphogen movement through gap junctions during early left-right patterning. This conceptual framework for modeling gap junctional signaling -- an epigenetic patterning mechanism of wide relevance in biological regulation -- suggests numerous experimental approaches in other patterning systems.
Figure 1. Proposed circumferential movement of morphogens through gap junctional communication (GJC) paths. A: In the animal pole tier of blastomeres of the 16–32 cell frog embryo, a ventral zone of junctional isolation (red line) exists around which the dorsal cells form a long-range cell path coupled by gap junctions. Both the barrier at the ventral midline and the open GJC path are required for normal left–right asymmetry (Levin and Mercola,1998). Green line, path of junctional communication; blue conduits, gap junctions. B: A similar situation exists with respect to the primitive streak in the early chick blastoderm (Levin and Mercola,1999). p.s., primitive streak. C: These observations suggested a model whereby asymmetry results from the net unidirectional movement of small molecule morphogens through the circumferential path; the gradient is schematized in blue:red in B and C, and the movement of the morphogen (schematized in yellow) is indicated by the red line in A. One possibility is that this movement is driven by left–right (LR) voltage differences generated across the zone of isolation by differential ion exchange of those cells with the outside medium. D–F: A maternal pool of serotonin is initially symmetrically (homogenously) distributed throughout the early embryo (D; Fukumoto et al.,2005b), but becomes present in a gradient (E), and eventually is restricted to one cell adjacent to the ventral embryonic midline (F). The spatial detection of serotonin was performed using immunohistochemistry (Levin,2004a).
Dependence of final serotonin concentration on voltage difference. The stationary serotonin concentration cS(x) (given by Eq. (4), final state that can be produced by the mechanism) across the Xenopus embryo is an exponential function and is given here for voltage differences between-10 mV and-40 mV. Medial circumferential length indicates position along the path from L to R border in an embryo roughly 1 mm in diameter (see Fig. 2).
Figure 4. Dependence of overall gradient gain on the electrical charge of morphogen. The serotonin right–left gain, which follows from the stationary concentration given in Eq. (4), exhibits an exponential dependence (shown here on a logarithmic plot) on both the electric charge of the morphogen molecule and the voltage difference. Thus, similar right–left gains (in different biological systems) may be obtained by different combinations of the morphogen charge and potential differences.
The dependence of overall gradient gain (left–right [L–R] steepness) on voltage difference. The stationary electrophoretic serotonin gain RS(x) within the embryo also follows an exponential function. It is shown here for the same voltage differences as in Figure 1. The morphogen gain measures how much more serotonin we expect to find within the embryo compared with the left side. Thus, RS(x = 0) = 1 at the left side, and increases toward the right side. The morphogen's right–left gain RS(x = L), that is, the maximum expected ratio of the serotonin concentration at the right side of the embryo with respect to the left side is approximately 2-fold, 5-fold, 10-fold, and so on, for a voltage difference of-10 mV,-20 mV, and-30 mV, respectively. Medial circumferential length indicates position along the path from L to R border in an embryo roughly 1 mm in diameter (see Fig. 2).
Figure 6. The temporal development of the gradient. The time-dependent development of the serotonin concentration in the Xenopus embryo, calculated from Eqs. (1,2), in a completely open path. A: Starting from a constant profile at t = 0 and a constant voltage difference of Δϕ =-20 mV, the exponential stationary profile is reached after approximately 1 hr. B: The serotonin right–left gain RS(L) as a function of time shows that the stationary right–left gain is reached after approximately 1 hr. A hypothetical smaller diffusion coefficient (smaller by a factor 1/√5 and intended to represent a five times greater morphogen mass), for comparison, leads to a slower generation of the morphogen gradient (almost 3 hr here). Medial circumferential length indicates position along the path from left (L) to right (R) border in an embryo roughly 1 mm in diameter (see Fig. 2).
Figure 7. The influence of gap junctions (GJs) on gradient development. The time-dependent development of the serotonin concentration in the Xenopus embryo. Gap junctional transport included by Eqs. (7–9) shows distinct effects. A: Starting from a homogenous profile at t = 0 (not shown here) and a constant voltage difference of Δϕ =-20 mV, the initial profiles appear ragged at each cellular interface such that the serotonin gradient across each individual cell is larger compared with the completely open embryo discussed in Figure 4, but evolves into the smooth stationary profile for later times. The number of GJs at each cellular interface here is NGJ = 105. B: The serotonin right–left gain RS(L) as function of time for different GJ densities. In comparison with a completely open syncytium, the GJ-mediated stationary profile is established on a slower time scale. This retardation depends strongly on the number of GJs at each cellular interface. C: The right–left gain is shown at t = 2 hr for different GJ densities.
Figure 8. Serotonin gradient across individual cells in the cell field. A,B: Space–time plot (A) and time course (B) of the individual cell serotonin gradients defined as concentration difference within each cell divided by the cell size vs. the embryonic serotonin gradient, defined here as difference of concentrations on the left and right side of the embryo divided by the embryo size. The individual cell gradients show heterogeneous behavior (compare, e.g., cell 8 [right side, gradient increases for all times] vs. cell 1 [left side; initial gradient increase but later decreasing]). The overall embryo gradient is initially smaller than all individual cell gradients but evolves into an intermediate value of all individual gradients.
Figure 7. The influence of gap junctions (GJs) on gradient development. The time-dependent development of the serotonin concentration in the Xenopus embryo. Gap junctional transport included by Eqs. (7–9) shows distinct effects. A: Starting from a homogenous profile at t = 0 (not shown here) and a constant voltage difference of Δϕ =-20 mV, the initial profiles appear ragged at each cellular interface such that the serotonin gradient across each individual cell is larger compared with the completely open embryo discussed in Figure 4, but evolves into the smooth stationary profile for later times. The number of GJs at each cellular interface here is NGJ = 105. B: The serotonin right–left gain RS(L) as function of time for different GJ densities. In comparison with a completely open syncytium, the GJ-mediated stationary profile is established on a slower time scale. This retardation depends strongly on the number of GJs at each cellular interface. C: The right–left gain is shown at t = 2 hr for different GJ densities.
Figure 8. Serotonin gradient across individual cells in the cell field. A,B: Space–time plot (A) and time course (B) of the individual cell serotonin gradients defined as concentration difference within each cell divided by the cell size vs. the embryonic serotonin gradient, defined here as difference of concentrations on the left and right side of the embryo divided by the embryo size. The individual cell gradients show heterogeneous behavior (compare, e.g., cell 8 [right side, gradient increases for all times] vs. cell 1 [left side; initial gradient increase but later decreasing]). The overall embryo gradient is initially smaller than all individual cell gradients but evolves into an intermediate value of all individual gradients.
