The Presence of Nuclear Cactus in the Early Drosophila Embryo May Extend the Dynamic Range of the Dorsal Gradient

In a developing embryo, the spatial distribution of a signaling molecule, or a morphogen gradient, has been hypothesized to carry positional information to pattern tissues. Recent measurements of morphogen distribution have allowed us to subject this hypothesis to rigorous physical testing. In the early Drosophila embryo, measurements of the morphogen Dorsal, which is a transcription factor responsible for initiating the earliest zygotic patterns along the dorsal-ventral axis, have revealed a gradient that is too narrow to pattern the entire axis. In this study, we use a mathematical model of Dorsal dynamics, fit to experimental data, to determine the ability of the Dorsal gradient to regulate gene expression across the entire dorsal-ventral axis. We found that two assumptions are required for the model to match experimental data in both Dorsal distribution and gene expression patterns. First, we assume that Cactus, an inhibitor that binds to Dorsal and prevents it from entering the nuclei, must itself be present in the nuclei. And second, we assume that fluorescence measurements of Dorsal reflect both free Dorsal and Cactus-bound Dorsal. Our model explains the dynamic behavior of the Dorsal gradient at lateral and dorsal positions of the embryo, the ability of Dorsal to regulate gene expression across the entire dorsal-ventral axis, and the robustness of gene expression to stochastic effects. Our results have a general implication for interpreting fluorescence-based measurements of signaling molecules.


S1 Text. Detailed description of model formulation
Original model formulation where The original model formulation features four equations representing nuclear dl, cytoplasmic dl, cytoplasmic dl/Cact complex and cytoplasmic Cact (Eqns 1-4, respectively). In these equations, C h s is the concentration of species s within nuclear/cytoplasmic compartment h, where s = {dl, n; dl, c; dc, c; Cact, c}.
The three cytoplasmic species are exchanged between neighboring compartments at a rate Γ s in proportion to the surface area of the two touching faces, A m (transport in the anterior-posterior direction is ignored, as the dl gradient is assumed to be approximately unchanging along this direction). Free dl is exchanged between the nucleus and cytoplasm at rates k out & k in in proportion to the surface area of each nucleus, A n . Cytoplasmic dl and Cact bind to form dl/Cact complex at a rate k b , and the unbinding rate of dl/Cact complex is B. Toll-mediated degradation of dl/Cact complex is goverened by a Gaussian function with parameters A and φ. Finally, Cact is produced uniformly at a rate P Cact and degraded at a rate k Deg .
In this model formulation, the nuclei are assumed well-mixed, and are initially empty at the start of each interphase. During interphase, the system is goverened by all four equations. At the end of interphase, the nuclei dissolve and nuclear dl is mixed with cytoplasmic dl within each compartment. During mitosis, the nuclei are undefined and the system is goverened by the three cytoplasmic equations, minus the terms for nuclear import/export. At the end of mitosis, the number of nuclei/compartments increases instantly, and the total concentration of each protein is interpolated into the new compartments. The concentration of nuclear dl starts at zero once again, and the next interphase begins. (See also [1].)

Updated Model Formulation
To allow for the presence of dl protein within nuclei at the start of interphase, we updated the model by removing the assumption that nuclei begin interphase empty. This updated model formulation is governed by six differential equations, now including equations for nuclear dl/Cact complex and nuclear Cact. At the beginning of each interphase, nuclear envelopes encapsulate a volume of cytoplasm as they reform. The concentration of each species is thus the same between the nucleus and cytoplasm before nuclear import/export dynamics take over.
The updated model consists of the following six differential equations: Non-dimensionalizing, we have: Cact for s = {dl, n; dl, c; dc, n; dc, c; Cact, n; Cact, c}.
The gene expression model equations take the general form: where f i is defined for each species i as the product of the appropriate on/off terms. To avoid discontinuities, we use a Hill function with n H = 100 to approximate the Heaviside step function: C n H /(θ n H + C n H ) for production and θ n H /(θ n H + C n H ) for repression:  N (0, 1) is a random number selected from the standard normal distribution. Values that fall below zero are set to zero.

Nuclear/cytoplasmic dimensions
For NCs 10-13, the volume and surface area of each nucleus are calculated assuming the nucleus is a sphere with constant radius r based on measurements in [2]. For nuclear cycle 14, the volume and surface area of each nucleus are calculated assuming the nucleus is a prolate spheroid with major and minor axes b and a, respectively, where b was approximated such that b = 2a. The surface area and volume parameters are normalized to the area and volume measurements of a nucleus at the end of NC14 (A 14 n ≈ 160µm 2 and V 14 n ≈ 190µm 3 ). The volume of the cytoplasm is simply the total volume of each compartment, V = H ( L /ni) 2 , where H = 25µm represents the constant height of the simulated array of compartments, L = 270µm is the length of the simulated array (i.e., the length of half the embryo's circumference), and n i is the number of nuclei in nuclear cycle i.

Derivation of η
To simulate the stochastic behavior of gene expression downstream of dl, we added artificial noise to the simulated dl nuclear concentration, δU nuc , in proportion to the square root of the nuclear dl concentration, U nuc , with η the proportionality constant. Here we provide a detailed derivation of η .
Berg and Purcell [3] argued that a microorganism could, at best, determine the concentration, c, of molecule X within a spherical volume of radius a with a fractional error of where T is the measurement timescale, and D is the diffusivity of molecule X (see also [4]). Thus, we assume that the relative fluctuations in concentration of dl at its binding site, C dl,n , are where D is the diffusivity of dl within the nucleus, a is the length of the DNA binding site, and T is an averaging time for the cell to measure this concentration [3,4]. If we apply non-dimensionalization to our dl nuclear concentration by substituting C dl,n = U nucCdl , we arrive at or, equivalently: Thus, the noise levels are δU nuc = η √ U nuc , and the proportionality constant is This is the formula employed in our gene expression simulations, in which, at each time step, a random level of Gaussian noise with a mean of zero and a standard deviation of δU nuc is added to the concentration of dl in each nucleus.
We estimate the following values for the physical parameters included in the calculation of η. First, the length of the dl binding site is 10 base pairs [5], meaning a ≈ 3 nm. Second, it seems reasonable to assumeC dl ≈ 1 − 10 nM ≈ 0.6 − 6 molc /µm 3 based on concentration measurements of Bicoid (FlyBase: FBgn0000166) [4]. Third, D is based on fluorescence correlation spectroscopy studies of Bicoid, in which its diffusivity was found to be roughly 1-10 µm 2 /s [6]. Finally, we also argue a timescale of approximately T ≈ 1 − 10 min is an appropriate averaging time for transcription during the early Drosophila embryo (see [7]). Therefore, using the mean value for each term in (22), we find that While this calculation does not prove that η = 0.2, it shows that our simulations are in accord with independent estimates of noise levels.

Estimation of T
For our estimate of T , we begin with the assumption that the averaging time for the concentration of dl to be read-out by a gene locus (ie, to result in a transcription decision) is related to the time between transcriptional bursts [8,9]. Recent work on transcriptional bursting events in the early Drosophila embryo suggests the timescale for bursting events is roughly 1-10 min [7,10,11].    Table. Dimensional quantities used in model formulation Figure S1: S1 Fig. Effect of noise on gene expression (Left to right) Increasing the nosie parameter, η from 0 to 1 shows that the slopes of the gene expression boundaries approach infinity at η = 0, and become very noise above η = 0.2. (Note: each run is an average of 10 runs for each parameter adjustment to reduce randomness in the plot due to noise. This comports with the experimental data, which are the average of 10+ embryos. The same is true for Fig. S2