Mathematical model of *zen* regulation

We explain our model using a rectangular embryo, where the x and y directions represent the AP and DV axes of the actual embryo. We assume that the AP and terminal signals are uniform along the DV axis. The rectangular case is presented here for simplicity, because it contains all of the essential features of our model, and leads to a simple algebraic expression that describes the boundary of *zen* expression. The same model applies for an embryo shaped as a prolate spheroid and was used to generate images in .

Let R denote a factor, such as Cic, that acts together with Dl to repress *zen*. The joint repressive action of Dl and R can be modeled by the product of Dl and R concentrations, Dl*R. In the model, *zen* is expressed when this product is below some threshold, denoted by Θ. Thus, the level of *zen* expression, denoted by Z, is given by *Z* (*x*, *y*) = *H* (Θ−*Dl*(*y*)×*R*(*x*)), where *H* (*u*) is the Heaviside step function: *H* (*u*)=1, when *u*> 0 and zero otherwise. The boundary of *zen* expression, denoted by *y*_{z}(*x*) is then given by the implicit function: Θ= *Dl*(*y*) × *R*(*x*). This equation can be related to the experimentally observed profiles of the Bcd, Dl, and Cic gradients.

We assume that repressor *R* is produced at a constant rate, denoted by *S*_{R}, and degraded via two parallel channels: constitutive degradation throughout the embryo and faster degradation that depends on the terminal signal. A steady state model for repressor level then becomes *S*_{R} = *k*_{c}R +*V* (*R*), where *k*_{c} is the rate constant of constitutive degradation, and *V* (*R*) is the rate law for the enzymatic degradation that depends on the terminal signal. We assume that the spatial pattern of active enzyme, denoted by *E*(*x*) (phosphorylated MAPK) is uniform along the DV axis. The spatial pattern of enzyme distribution *E*(*x*) is given by the product of the amplitude *E*_{0} and a symmetric function *f*_{E}(*x*) that is equal to one at the poles and close to zero in the middle of embryo.

We assume that degradation of *R* follows Michaelis-Menten kinetics, with constants *k*_{cat} and *K*_{M}. We also assume that Bcd competitively inhibits MAPK-dependent repressor degradation. This leads to the following expression for the rate of signal-induced repressor degradation: *V* (*R*) = *k*_{cat}E (*x*)*R*(*x*) / (*K*_{M} + *R*(*x*) + *K*_{M}B(*x*) / *K*_{I}), where *K*_{I} is the equilibrium constant of enzyme-Bcd interaction.

To simplify the algebra, we assume that the enzyme is not saturated by

*R*. Under this assumption,

*V*(

*R*) =

*k*_{cat}E(

*x*)

*R* /

*K*_{M} (1+

*B*(

*x*) /

*K*_{I}). Substituting this into the mass balance for repressor levels we get the following expression for the spatial pattern of

*R*:

where

*α* and

*β* are defined as:

*α*=

*k*_{cat}E_{0}/

*k*_{c}K_{M},

*β*=

*B*_{0}/

*K*_{I}. These parameters can be estimated from the wild type pattern of Cic downregulation: At the posterior pole, where

*f*_{E}(

*x*) =1, but the concentration of anteriorly localized Bcd is zero (

*f*_{B}(

*x*) = 0), Cic is downregulated to 10% of its level in the midbody of the embryo, where

*f*_{E}(

*x*) =0. From this result, we find that

*α* ≈

*9*. At the same time, at the anterior of the embryo, where

*f*_{B}(

*u*) =1, Cic is downregulated only to ~50% of its midbody level. Based on this result, and on the estimate for

*α*, we get that

*β* ≈

*4*. Combining these estimates with the shape of the Bcd gradient and the shape of the terminal signal, we can predict how

*R* is distributed throughout the AP axis of the embryo.

The equation for the boundary of

*zen* expression then takes the following form:

where

*D*_{0} is the amplitude of the nuclear Dl gradient and

*f*_{D} (

*y*) is the shape that characterizes its distribution along the embryo.

Introducing one more dimensionless group

*γ* Θ

*k*_{c}/(

*S* ×

*D*_{0}), we get the following equation for the

*zen* expression boundary:

The remaining parameter of the model,

*γ*, can be estimated from the location of the

*zen* boundary in the midbody region of the embryo, where

*f*_{E}(

*x*) ≈ 0, which implies that

*γ* =

*f*_{D} (

*y*_{Z,m}), where

*y*_{Z,m} denotes the position of the

*zen* boundary at the midbody region of the embryo. Based on our previous imaging results (

Chung et al., 2011;

Kanodia et al., 2009), we estimate that

*γ* ≈

*0.1*.

Putting everything together we get the following equation for the expression boundary of the wild type pattern:

Note that the values of the three dimensionless groups in the model were obtained from the asymmetries of the wild type pattern of Cic downregulation, the location of the *zen* expression boundary in the midbody region of the embryo, and the spatial distribution of the nuclear Dl gradient in the midbody region.

We can now combine the values of *α*, *β*, and *γ* with the empirically determined distributions for the patterning signals, *f*_{D}(*y*), *f*_{E}(*x*), and *f*_{B}(*x*), to plot the wild type pattern of *zen* expression. This model predicts how the *zen* expression boundary “bends” in response to variations in the levels of anterior, terminal, and dorsoventral signals. For example, removing Bcd makes MAPK more available for *R*, lowering a co-repressor that acts together with Dl in *zen* repression, and results in ectopic *zen* at the anterior pole ().

HIGHLIGHTS- MAPK substrates compete for access to MAPK in the early
*Drosophila* embryo - Anterior substrates promote Capicua action by blocking MAPK-dependent degradation
- Substrate competition explains gene expression patterns with complex AP/DV polarity