# Antibody Modelling – Immunoinformatics

In the battles with infective diseases, warriors fight to protect their country - your body. They are the (un)known warriors, antibodies. The specificity of an antibody is determined by its paratope (antigen-binding site), which is mainly made up of complementarity-determining regions (CDRs). There are 3 CDRs on each of the two polypeptide chains (heavy and light chains) … Continue reading Antibody Modelling – Immunoinformatics

# Soft Tissue Mechanics – Constitutive Law

Constitutive laws describe material properties (strains and stresses). Since soft tissues are composed of fibres, which are not aligned, the angles at which the fibres are oriented have an impact on their mechanical properties. Here we consider the contributions from i) along the fibre $latex \mathbf{f}$; ii) along the sheet $latex \mathbf{s}$; and iii) in … Continue reading Soft Tissue Mechanics – Constitutive Law

# Antibody i-Patch: Paratope prediction

Paratope: Antibody i-Patch: Antibody contact residues prediction Binding likelihood score of each residue $latex a$: $latex p_a = \frac{\frac{f^{con}_a}{f^{non}_a}}{\frac{\sum f^{con}_b}{\sum f^{non}_b}}$ where $latex p_a$ is the propensity to be in contact, $latex f^{con}_a$ and $latex f^{non}_a$ are the surface frequency of residue $latex a$ to be in contact ($latex con$) and not to be in … Continue reading Antibody i-Patch: Paratope prediction

# Finite Element

How can you tell whether a bridge can bear the load of all the vehicles on it? Yours skin is stretchy - until a point it breaks. But how far can you stretch your skin before it breaks? In cases where you can't repeatedly build a million bridges nor stretch your skin a million times … Continue reading Finite Element

# True stress, and relative stress (Finite Deformation)

I mean, stress tensor. Finite element analysis was, in its early days, proposed for engineering solutions to dam constructions - by discretising a continuous domain into smaller elements and approximating the displacements of each elements upon loading. A major advantage of FEM over FDM is its ability to handle complex geometry: in FEM, the domain … Continue reading True stress, and relative stress (Finite Deformation)

# Finite Element Method (Weak form)

In order to apply Finite Element Method to solve PDEs, we need to convert the PDEs into an equivalent "weak form". Example: Axially loaded Elastic Bar Strong Form: $latex \frac{d}{dx}\big(AE\frac{du}{dx}\big)+b=0$ with boundary conditions: at $latex x=0$, traction/force per unit area $latex \bar{t}$, is prescribed: $latex \sigma(x=0) = \big(E\frac{du}{dx}\big)_{x=0}=\frac{p(0)}{A(0)}=-\bar{t}$ at $latex x=l$, displacement $latex \bar{u}$ is … Continue reading Finite Element Method (Weak form)

# Finite Element Method (Simplex, Linear Elastic)

Finite Element method is a numerical method, rather than analytic method of solving PDEs. Interpolation function Defines the type of polynomials used approximate the real solution Higher order polynomials have some flexibilities to approximate the actual solution more accurately However it requires more information and operations to compute a higher order element. For example, in 1D, a … Continue reading Finite Element Method (Simplex, Linear Elastic)

# Finite Element Method (Simplest case)

Motivation: Approximating solutions to ODE/PDE in a complex geometry Steps: Domain Discretisation Derivation of Equations Assemble Element Equations Boundary Conditions Solution Case: Assume a cylinder with radius $latex R$ and length $latex L$. Changes in Temperature: $latex \hat{\theta} = (T - T_{env})$ Position of node: $latex \eta = \frac{x}{L}$ Constant: $latex \mu^2 = L^2 \frac{2h}{kR}$ … Continue reading Finite Element Method (Simplest case)

# An alternative perspective to differential equations systems

Foreword: In biological system modelling, often we are faced with a (huge) set of differential equations. They describe physical phenomena and are therefore relatively easy to construct and inspect if the system was constructed in the right way. However, solving these are computationally taxing and there is no way to predict the outcome just by … Continue reading An alternative perspective to differential equations systems