Complex biological products, such as those used to treat various forms of cancer, are typically produced by mammalian cells in bioreactors. The most important class of such biological medicines is proteins. These typically bind to sugars (glycans) in a process known as glycosylation, creating glycoproteins, which are more stable and effective medicines. The glycans are large polymers that are formed by a long sequence of enzyme catalysed reactions. This sequence is not always completed, thus leading to a heterogeneous glycoprotein distribution. A better comprehension of this distribution could lead to more efficient production of high quality drugs. To understand how the manufacturing process can affect the extent of glycosylation of protein, a non-linear ODE model of glycoprotein production is developed which describes the bioreactor configuration as well as the protein production and glycosylation reactions within the cell. The entire system evolves eventually to a stable steady state. The earlier evolution is critical however, as the amount of product produced and its quality varies over time. The model is considered as two coupled systems: the bioreactor submodel and the glycosylation submodel. To investigate the early time evolution within the bioreactor submodel, analytical and numerical properties are derived using matched asymptotic expansions and a finite difference scheme for a range of initial conditions. This leads to qualitatively different regimes for aglycosylated protein production, which affect the glycosylation submodel. The discrete glycoprotein distribution is approximated as continuous and written as a first-order PDE, with good agreement between the discrete and continuous models. The PDE is found to admit shocks, but the existence of these shocks is dependent on the early time evolution within the bioreactor submodel and leads to higher levels of glycosylation at early time. This suggests that changing the bioreactor configuration can lead to higher quality product at certain times.

This paper re-examines the problem of the flow of a fluid of finite depth over two Gaussian-shaped obstructions on the stream bed. A weakly nonlinear analysis in the form of the Korteweg–de Vries equation is used to compare with the results of the fully nonlinear problem. The main focus is to find waveless subcritical solutions, and contours showing the obstruction height and separation values that result in waveless solutions are found for different Froude numbers and different obstruction widths.

We consider the Gierer–Meinhardt system with precursor inhomogeneity and two small diffusivities in an interval

$$\begin{equation*} \left\{ \begin{array}{ll} A_t=\epsilon^2 A''- \mu(x) A+\frac{A^2}{H}, &x\in(-1, 1),\,t>0,\\[3mm] \tau H_t=D H'' -H+ A^2, & x\in (-1, 1),\,t>0,\\[3mm] A' (-1)= A' (1)= H' (-1) = H' (1) =0, \end{array} \right. \end{equation*}$$

$$\begin{equation*}\mbox{where } \quad 0<\epsilon \ll\sqrt{D}\ll 1, \quad \end{equation*}$$

$$\begin{equation*} \tau\geq 0 \mbox{ and $\tau$ is independent of $\epsilon$. } \end{equation*}$$

$O(\sqrt{D})$

The gradient flow structure of the model introduced in Cermelli & Gurtin (1999, The motion of screw dislocations in crystalline materials undergoing antiplane shear: glide, cross-slip, fine cross-slip. Arch. Rational Mech. Anal.148(1), 3–52) for the dynamics of screw dislocations is investigated by means of a generalised minimising-movements scheme approach. The assumption of a finite number of available glide directions, together with the “maximal dissipation criterion” that governs the equations of motion, results into solving a differential inclusion rather than an ODE. This paper addresses how the model in Cermelli & Gurtin is connected to a time-discrete evolution scheme which explicitly confines dislocations to move at each time step along a single glide direction. It is proved that the time-continuous model in Cermelli & Gurtin is the limit of these time-discrete minimising-movement schemes when the time step converges to 0. The study presented here is a first step towards a generalisation of standard gradient flow theory that allows for dissipations which cannot be described by a metric.

We study a system of two coupled parabolic equations that models the spreading of a drop of an insoluble surfactant on a thin liquid film. Unlike the previously known results, the surface diffusion coefficient is not assumed constant and depends on the surfactant concentration. We obtain sufficient conditions for finite speed of support propagation and for waiting-time phenomenon by application of an extension of Stampacchia's lemma for a system of functional equations.

Inviscid irrotational flow around a pair of coaxial disks is considered in the limit in which the distance 2h between the disks is small compared to their radius a. The disks have zero thickness and accelerate away from one another along their common axis. The added mass M of each accelerating disk is increased by the presence of the other disk. Analytic predictions are obtained when h/a ≪ 1, with M ~ πa/(8h)-ln(h/a)/2+0.77875+. . . The term O(a/h) can be obtained by means of an inviscid analysis of approximately unidirectional flow within the gap between the disks, but the correction terms have not been reported previously. The irrotational flow problem satisfies Neumann boundary conditions on the surface of the disks, but is otherwise analogous to the Dirichlet problem of the capacitance of a pair of charged disks, which has been the subject of much study and controversy.