In such cases, we can still take advantage of the visualisation power of potential landscapes by approximating the true potential using a numerical method.
Then analytical solutions of eq 5.59 of the form 5.61are possible, for example, by approximating the effective potential as a double harmonic oscillator in the R and Q coordinates.
Most biomolecular simulations account for the presence of the native (cellular) environment either explicitly, for example by inclusion of water molecules and ions, or implicitly, by approximating the mean force (Potential of Mean Force, PMF) exerted by the external media on the biomolecule.
In the texture-mapping procedure, we begin by approximating the target surface with an approximating cylinder.
The main assumption is that the incomplete empirical probability distribution (determined by occurrence data) can be approximated with a probability distribution of maximum entropy (the Maxent distribution) subject to certain environmental constraints, and that this distribution approximates the potential geographic distribution of the group of interest [56].
For this reason we used an indirect method based on the estimate of the free energy profile for the permeation process [ 15, 16], which can be approximated by the potential of mean force (PMF) [ 17] for calcein permeation through a homomeric hCx26wt hemichannel.
For the approach which approximates the floating potential by considering Ω F to be a dielectric with a high relative permittivity ε F, we have to solve the following system of linear equations ( ε D S D D, h D + ε 0 S D D, h 0 ε 0 S D F, h 0 ε 0 S F D, h 0 ε F S F F, h F + ε 0 S F F, h 0 ) ( φ ̲ D φ ̲ F ) = ( f ̲ D f ̲ F ). (18).
For the approach which approximates the floating potential by considering Ω F as a dielectric with a high relative permittivity ε F, the corresponding system reads ( V E E, h V E F, h V E D, h K ˜ F E, h ⊤ 1 2 ε F + ε 0 ε F − ε 0 M ˜ h + K ˜ F F, h ⊤ K ˜ F D, h ⊤ K ˜ D E, h ⊤ K ˜ D F, h ⊤ 1 2 ε D + ε 0 ε D − ε 0 M ˜ h + K ˜ D D, h ⊤ ) ( w ̲ E w ̲ F w ̲ D ) = ( f ̲ E 0 ̲ 0 ̲ ).
In the general case, one can approximate the potential (Q x)) by the sequence of potentials Q_{beta}(x) = begin{cases} Q x), &0 le x lebeta, 0,& x > beta, end{cases} and use the technique from [5] (see Lemma 2.4.1).
