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In vertebrate limb, a group of specialized epithelial cells called Apical Ectodermal Ridge (AER) form at the boundary of dorsal and ventral limb ectoderm. Recent experiments suggest that AER forms at the boundary of Fringe expressing and Fringe non-expressing cells by a specific type of receptor-ligand interaction called as inductive signaling, involving the transmembrane proteins Notch, Serrate and Delta. Experiments conducted on Drosophila wing disc have shown that Fringe inhibits the binding ability of Serrate ligand to Notch and enhances that of Delta to Notch. Although several of the signaling elements have been identified experimentally, it remains unclear how the inter-cellular interactions can give rise to such a boundary of specialized cells. Here we present an ordinary differential equation (ODE) model involving Delta→Notch and Serrate→Notch interactions between juxtaposed Fringe expressing and Fringe nonexpressing cells. When simulated in a compartmentalized set up, this model gives rise to high Notch levels at the boundary of Fringe expressing and Fringe non-expressing cells.

Notch signaling is one of the highly conserved signaling pathways in animal kingdom and plays a vital role in determining the fate of developing tissues [

Notch signaling in developing tissues can be categorized into three types: 1) Lateral inhibition: Here equivalent cells in a population acquire different fate due to Notch signaling resulting in one cell with high Notch and the neighboring cell with low Notch forming a checker board pattern; 2) Lineage decisions: An asymmetric cell division occurs where cell progenies acquire different levels of Notch resulting in differentiated cell types, and 3) Induction: Here specialized cells are formed at the boundary of two non-equivalent cell populations that express different genes [2,3]. Inductive signaling of Notch plays a vital role in Drosophila wing disc formation and limb formation in vertebrates [4,5]. In chick embryos, at HH (Hamburger-Hamilton) [

expressed in the dorsal ectoderm influences the dorsal fate of the ectoderm and the underlying mesenchyme [7,8]. Engrailed En-1 is expressed in the ventral ectoderm and has been shown to influence the ventral cell fate [

Recent experimental work by Rodriguez-Esteban et al. [

Although experiments in Drosophila have shed light on several elements of the signaling mechanism, a complete understanding of this inductive signaling is still unclear. One way of verifying the core signaling is through mathematical modeling involving both interand intra-cellular signaling. In this short report, we propose a mathematical model using ordinary differential equation (ODE) to describe the inductive Notch signaling leading to boundary formation. In order to couple the Notch signaling at the inter-cellular level, we use multiple compartment ODEs. Using our model we show the formation of boundary in 1D dimensional arrangement of cells which qualitatively agrees with the higher Notch levels at the AER in chick limb. We finally discuss the potential application of the model and its future extensions in into 2D and 3D spatial scales.

To build the model we first consider two non-equivalent cell populations of the same cell type juxtaposed in a 1D line: Fringe expressing (dorsal) (2 pink cells in

• Dorsal cells express Serrate and Notch.

• Ventral cells express Notch and Delta.

• In the dorsal cells, activation of Notch through Serrate is prevented by Fringe: Fringe modifies the Notch ligand through glycosylation and decreases Notch’s affinity to Serrate.

• In the ventral cells activation of Notch through Delta is not strong in the absence of Fringe.

• At the boundary, dorsal Serrate is able to activate the Notch in ventral cells without being inhibited by Fringe and ventral Delta is able to activate the Notch in the dorsal cells which is enhanced by Fringe.

• The resulting high Notch activity at the boundary, in turn activates the transcription of the ligands Serrate and Delta possibly forming a positive feedback loop [

• The positive feedback loop sustains the high Notch activity as well as the high ligand activity at the boundary.

The above mechanism is illustrated in

Here N_{i}, D_{i}, S_{i} and F_{i} represent levels of active Notch, Delta, Serrate and Fringe proteins in cell i, f and g are monotonously increasing and decreasing functions respectively. f and g vary between 0 and 1. k_{syn} and k_{deg} are synthesis rate and degradation rate constant of each of the species. Z represents the factor responsible for the differential expression of Serrate and Delta in dorsal and ventral compartments respectively. Z can be considered similar to a cofactor required for the transcription of a specific gene. There is experimental evidence for restriction of Serrate expression in dorsal compartment in drosophila, however, the expression of Delta is unclear [_{s} = 1 for dorsal cells and Z_{s} = 0 for ventral cells. There is evidence of complimentary expression of Serrate and Delta in chick neural tube [_{nc} indicates the activity of Delta in neighboring cells. In a 1D line of cells

and in a 2D arrangement, it is the average taken over the immediate neighbors and given as:

The same principle applies for Serrate also. We do not consider cell division since cell cycle times are much greater than the time scale of the inter-cellular dynamics and hence we assume it does not have an effect.

The proposed model can be reduced to a one variable system by making the variables Serrate and Delta constants along with Fringe. We consider n cells arranged in a line in a 1D representation (

where the factor Φ_{i} represents all the terms in the parenthesis in Equation (1). Since Serrate (S) and Delta (D) are constants, the factor Φ_{i} will also be a constant depending upon the value of S and D in the neighboring cells. This results in a simple stable system with steady states N_{i} = 0, K_{i} and solution:

Here

The simulation of various solution curves of N(t) is given in _{i}, which implies that the steady state level of Notch depends on the ratio of its synthesis and degradation rate. The parameter K_{i} depends on the Serrate and Delta activities of the neighboring cells. For each cell, the value of Φ_{i} is different. This results in a series of coupled system with different K_{i} s, eventually giving rise to different steady state in each of the cells. Although the dynamics of the single cell is less interesting, when coupled as a four cell system, the two cells at the boundary results in a steady state that has a higher Notch activity when compared to the cells in the either side of the line of cells. We illustrate this in the following section.

The three-variable model given in Equations (1)-(3) is more complex as it involves the dynamics of Serrate and Delta. The most important aspect of this model is that a positive feedback is induced by the active Notch over the synthesis of the ligands Serrate and Delta. Unlike the previously known model of lateral inhibition [

In order to construct a multi-compartment ODE model, we consider the model in Equations (1)-(3) where we have 4 compartments representing four cells i = 1, 2, 3, 4. The initial conditions are set such that compartments 1 and 2 have non-zero levels of Fringe, Serrate and Notch, while compartments 3 and 4 have non-zero levels of Delta and Notch. In each compartment

and

(See _{1} ··· N_{4}, S_{1} ··· S_{4}, and D_{1} ··· D_{4}. We can see that in _{1} and N_{4} are zero where as N_{2} and N_{3} which are at the boundary, are high. Similarly the concentrations of D_{1}, D_{2} and D_{4} are zero while D_{3} is high (_{1}, S_{3} and S_{4} are zero while S_{2} has a non-zero high value (

In the recent years, significant progress has been made in understanding both the origin and molecular nature of the signals controlling patterning of the dorsoventral limb axis and AER formation. Inductive signaling of Notch has not been well understood despite its critical role in Drosophila wing disc as well as vertebrate limb formation. In addition to this, differential expression of fng and serrate genes has been a hurdle for mathematical modeling efforts. In this short report, we propose an ODE based compartmental mathematical model to describe inductive Notch signaling involved in the boundary formation at the dorsoventral limb axis. Our model is phenomenological and hence does not involve any Notch-related biochemical reactions. This qualitative approach allows us to derive the following conclusions:

• Boundary of specialized high Notch-expressing cells is formed due to the interaction of two cell populations with differential gene expressions: In real biological systems, differential gene expression patterns are programmed in the developmental protocols and hence they need to be considered as such. To interpret this differential gene expression mathematically, we suppressed the dynamics of the relevant variable in the respective compartments. However, this approximation limits the application of global analysis of the model.

• The positive feedback loop at the boundary cells further maintains high Notch levels by activating transcription of more Serrate and Delta: In the experiments on chick limb AER formation, initially Serrate expression is observed throughout the dorsal side and then restricts only to the AER [9,12]. To simulate this observation, the model requires a positive feedback loop from Notch to Serrate and Delta formation, eventually creating a boundary which expresses high Serrate and Delta in addition to Notch. However, this result is left to be shown experimentally.

• This form of model can only account for fine-grained patterns of cell specialization: Our model explains the interactions only between one nearest neighbor. However, there may be long range interactions, which are not accounted in this model.

This is the first attempt to model inductive Notch signaling giving rise to boundary formation in developing tissues. An added advantage of this approach is that this can be extended into 2D arrangement of cells as well as into any agent based modeling approaches potentially leading to a multi-scale model. We are currently making efforts to incorporate this model in a cell-based modeling environment in a 2D and 3D spatial arrangement as well as parameter search that can show this behavior. Our model is the first to represent differential gene expression mathematically and is able to simulate the boundary formation. Our model has some limitations such as 1) absence of dynamics of Fringe and 2) absence of biochemical reactions involving Notch-Delta ligand formation. Nevertheless, this model presents a versatile framework on which further extensive models can be built.

We would like to acknowledge the EPA grant—The Texas-Indiana Virtual STAR Center; Data-Generating in vitro and in silico Models of Developmental Toxicity in Embryonic Stem Cells and Zebrafish.