Calculate equilibrium chemical activities of species from the affinities of formation of the species at unit activity.
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list, output from
character or numeric, how to balance the transformations
numeric, which species to include
logical, normalize the molar formulas of species by the balancing coefficients?
logical, report results for the normalized formulas?
numeric, affinities of formation reactions excluding species contribution
numeric, number of moles of conserved component in the formation reactions of the species of interest
numeric, logarithm of total activity of balanced quantity
character, equilibration method to use
equilibrate calculates the chemical activities of species in metastable equilibrium, for constant temperature, pressure and chemical activities of basis species, using specified balancing constraints on reactions between species.
It takes as input
aout, the output from
affinity, which may be calculated from a multidimensional grid of conditions.
The equilibrium chemical activities of species are calculated using either the
equil.boltzmann functions, the latter only if the balance is on one mole of species.
aout contains the chemical affinities of formation reactions of each species of interest.
equilibrate needs to be provided constraints on how to balance the reactions representing transformations between the species.
balance indicates the balancing constraints, according to the following scheme:
name of basis species: balance on this basis species
length: balance on length of proteins
1: balance on one mole of species
numeric vector: user-defined constraints
The default value of NULL for
balance indicates to use the coefficients on the basis species that is present (i.e. with non-zero coefficients) in all formation reactions, or if that fails, to set the balance to 1.
However, if all the species (as listed in code
aout$species) are proteins (have an underscore character in their names), the default value of NULL for
balance indicates to use length as the balance.
NOTE: The summation of activities assumes an ideal system, so ‘molality’ is equivalent to ‘activity’ here.
loga.balance gives the logarithm of the total activity of
balance (which is total activity of species for 1 or total activity of amino acid residue-equivalents for length).
loga.balance is missing, its value is taken from the activities of species listed in
aout; this default is usually the desired operation.
normalize if TRUE indicates to normalize the molar formulas of species by the balance coefficients.
This operation is intended for systems of proteins, whose conventional formulas are much larger than the basis speices.
The normalization also applies to the balancing coefficients, which as a result consist of 1s.
After normalization and equilibration, the equilibrium activities are then re-scaled (for the original formulas of the species), unless
as.residue is TRUE.
equil.boltzmann is used to calculate the equilibrium activities if
balance is 1 (or when
as.residue is TRUE), otherwise
equil.reaction is called.
The default behavior can be overriden by specifying either boltzmann or reaction in
equil.reaction may be needed for systems with huge (negative or positive) affinities, where
equil.boltzmann produces a NaN result.
ispecies can be supplied to identify a subset of the species to include in the calculation.
equil.boltzmann each return a list with dimensions and length equal to those of
Astar, giving the
log10 of the equilibrium activities of the species of interest.
equilibrate returns a list, containing first the values in
aout, to which are appended
m.balance (the balancing coefficients if
normalize is TRUE, a vector of 1s otherwise),
n.balance (the balancing coefficients if
normalize is FALSE, a vector of 1s otherwise),
loga.equil (the calculated equilibrium activities of the species).
The input values to
equil.boltzmann are in a list,
Astar, all elements of the list having the same dimensions; they can be vectors, matrices, or higher-dimensionsal arrays.
Each list element contains the chemical affinities of the formation reactions of one of the species of interest (in dimensionless base-10 units, i.e. A/2.303RT), calculated at unit activity of the species of interest.
The equilibrium activities (in base-10 log units) of the species of interest returned by either function satisfy the constraints that 1) the final chemical affinities of the formation reactions of the species are all equal and 2) the total activity of the conserved component is equal to (
The first constraint does not impose a complete equilibrium, where the affinities of the formation reactions are all equal to zero, but allows for a metastable equilibrium, where the affinities of the formation reactions are equal to each other.
equil.reaction (the algorithm described in Dick, 2008 and the only one available prior to CHNOSZ_0.8), the calculations of relative abundances of species are based on a solving a system of equations representing the two constraints stated above.
A close approximation of the solution is found using
Prior to CHNOSZ_0.9-9, the values in the
Astar were used to generate initial guesses of the logarithms of activities of species; values of
loga.balance that were hugely different from these guesses could generate errors from
uniroot such as "f() values at end points not of opposite sign".
Now (from version 0.9-9), a more flexible method for generating guesses is in place.
equil.boltzmann (algorithm available beginning with CHNOSZ_0.8), the chemical activities of species are calculated using the Boltzmann distribution.
This calculation is faster than the algorithm of
equil.reaction, but is limited to systems where the transformations are all balanced on one mole of species.
equil.boltzmann is called with
balance other than 1, it stops with an error.
Dick, J. M. (2008) Calculation of the relative metastabilities of proteins using the CHNOSZ software package. Geochem. Trans. 9:10. https://doi.org/10.1186/1467-4866-9-10
diagram has examples of using
equilibrate to make equilibrium activity diagrams.
revisit can be used to perform further analysis of the equilibrium activities.
palply is used by both
equil.boltzmann to parallelize intensive parts of the calculations.
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## equilibrium in a simple system: ## ionization of carbonic acid basis("CHNOS+") species(c("CO2", "HCO3-", "CO3-2")) # set unit activity of the species (0 = log10(1)) species(1:3, 0) # calculate Astar (for unit activity) res <- 100 Astar <- affinity(pH=c(0, 14, res))$values # the logarithms of activity for a total activity # of the balanced quantity (C or CO2) equal to 0.001 loga.boltz <- equil.boltzmann(Astar, c(1, 1, 1), 0.001) # calculated another way loga.react <- equil.reaction(Astar, c(1, 1, 1), 0.001) # probably close enough for most purposes stopifnot(all.equal(loga.boltz, loga.react)) # the first ionization constant (pKa) ipKa <- which.min(abs(loga.boltz[] - loga.boltz[])) pKa.equil <- seq(0, 14, length.out=res)[ipKa] # calculate logK directly logK <- subcrt(c("CO2","H2O","HCO3-","H+"), c(-1, -1, 1, 1), T=25)$out$logK # we could decrease tolerance here by increasing res stopifnot(all.equal(pKa.equil, -logK, tolerance=1e-2))
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