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 yz(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 SR, 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 SR = kcR +V (R), where kc 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 E0 and a symmetric function fE(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 kcat and KM. 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) = kcatE (x)R(x) / (KM + R(x) + KMB(x) / KI), where KI 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
) = kcatE
) / KI
). Substituting this into the mass balance for repressor levels we get the following expression for the spatial pattern of R
are defined as: α
. These parameters can be estimated from the wild type pattern of Cic downregulation: At the posterior pole, where fE
) =1, but the concentration of anteriorly localized Bcd is zero (fB
) = 0), Cic is downregulated to 10% of its level in the midbody of the embryo, where fE
) =0. From this result, we find that α
. At the same time, at the anterior of the embryo, where fB
) =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
. 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:
is the amplitude of the nuclear Dl gradient and fD
) is the shape that characterizes its distribution along the embryo.
Introducing one more dimensionless group γ
), we get the following equation for the zen
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 fE
) ≈ 0, which implies that γ
), where yZ,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
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, fD(y), fE(x), and fB(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 ().
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