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Modelling

A model may be at various level of detail. The properties of a material or of a chemical process depend eventually on atomic and molecular properties. Sometimes it is actually reasonable to design a model on this level of description. In a molecular simulation, each atom is regarded individually and the forces between atoms are expressed as functions of distances between atoms and angles between molecular bonds. A box of molecules may be simulated in for instance Molecular Dynamics Simulations and Monte Carlo Simulations. An example below will show how the solubility of a drug may be calculated by Monte Carlo Simulations.

Sometimes a process of interest may be well described by macroscopic properties. Diffusion is such an examples. According to Fick's law, the flux of matter is proportional to the gradient in concentration. A model including diffusion normally leads to a Partial Differential Equation (PDE). Models for diffusion may for instance be used for calculating permeabilities of membranes and thin films. Heat conduction due to a non-uniform distribution of temperature is analogous to diffusion.

Our models often lead to molecular simulation or systems of partial or ordinary differential equations. But many other techniques may be required. For instance, construction of phase diagrams involves solving systems of non-linear equations. Polymer networks may successfully be investigated by Graph Theory. An example of this will be given below.

Due to confidentiality to our clients, most of our research cannot be presented here. Sometimes, however, our clients want to publish the results to the international community and then we can describe our work publicly.


Drug solubility

Consider a situation of mixing water and an excess amount of powder of a drug substance of amorphous structure. The temperature and pH are fixed. The drug may be an acid or a base but the pH is chosen so that the drug is almost always in its neutral form in the solution. Molecules will be released from the powder until saturation is reached. When no more drug can be dissolved from the solid powder the concentration of drug in the aqueous phase is called the solubility of the drug. The solubility is an important parameter in pharmaceutics. A substance with high solubility may have a high uptake in the body and hence be a good candidate for a product.


Drug dissolved (small white circles) in water (blue). Excess solid amorphous drug (white bottom) is present.

What determines the solubility? At equilibrium, i.e. saturation, the drug molecules are equally "comfortable" in the solution as in the solid material. We use the measure chemical potential for how comfortable a drug molecule is. The lower chemical potential, the more comfartable is the molecule. The chemical potential in the solid material is fixed but for a solution it varies with concentration. If the solution is very diluted, the chemical potential is low so molecules prefer to leave the solid for the solution. But the chemical potential normally increases with concentration so eventually, when the solution is saturated, the chemical potential is the same in both phases.

In order to obtain the solubility from simulations we need a way to calculate the fixed chemical potential in the solid material and the chemical potential of the drug in the solution as a function of concentration.

Let us start with the solution. By definition, the chemical potential of the drug in water is the free energy required to insert one extra drug molecule in the solution. The more strongly a drug molecule interacts with water, the less free energy is required for the insertion. Thus strong drug-water interaction favours high solubility. However, there is also an entropic part that must be considered.

In a Free Energy Perturbation (FEP) Monte Carlo simulation, a drug molecule can slowly be inserted in a box of water molecule. The interaction between the atoms is decribed by van der Waals forces, electrostatic forces and covalent bond forces. The simulation will give the chemical potential at infinite dilution. By adding the chemical potential term for ideal mixing, RTlnC, where R = 8.31 J/molK, T is the temperature and C the concentation, the chemical potential as function of concentration is obtained.



When a drug molecule is inserted into a box of water molcules it first takes a small shape. As the system is at constant presure, the volume of the box will adjust to give the new molecule room.

A FEP simulation of solid amorphous drug material is more difficult as the mobility of the molecules at room temperature may be very low. A trick is then to heat the solid material to for instance 400C. The mobility is then higher and a molecule may be instered. The chemical potential obtained 400C must then be transformed into the value corresponding to room temperature. That may be done by the Gibbs-Helmholtz equation in the method of Thermdynamic Integration. In the figure below the point of intersection between chemical potentials for drug in water and drug in solid is shown.


The intersection between chemical potential for drug in solution
and drug in solid yields the solubility.



Membrane permeability

Imagine a membrane separating two aqueous solutions of a substance. The concentration of the substance is higher on one of the sides. By diffusion, a net flow of substance will take place from the high concentration side to the other side.

The flow of substance is given by

F = P A ΔC.

where A is the corss-section area of the membrane, ΔC the difference in concentration between the two sides. The higher the permeability, P, the higher is the flow through the membrane. Let us say that the membrane can be considered as a network of N channels leading between the sides. Let us also say that each channel going from node i to j has the permeability p(ij). In order to make an analogy to a network of electric resistances we define the resistance to flow in a channel as r(ij) = 1/p(ij).



The channels enabling permeation through a membrane.
To create a closed "curcuit", an artifical return channel
with zero resistance is added.

To calculate the total flow through the network we add a channel outside the membrane with zero resistance. Then we can apply Kirchhoff's laws for electrical circuits. Translated to a flow-concentration system they state that the net flow to each node must be zero and that the net change in concentration when moving around a closed loop is zero. Setting up these equations for all nodes and all closed loops will give as a system of linear equations. Its solution yields all the flows through channels and subsequantly the total flow and the total permeability.

The complication with this model is that the closed loops that should be considered in Kirchhoff's laws must be linearly independent. In the figure below there are three closed loops drawn, but there are only two linearly independent ones. A third loop can be expressed as a linear combination of the two others. We have developed an algorithm that quickly and robustly finds a complete set of linearly independent closed loops.


The three closed loops are not linearly independent.