CHNOSZ (version 1.1.0)

equilibrate: Equilibrium Chemical Activities of Species

Description

Calculate equilibrium chemical activities of species from the affinities of formation of the species at unit activity.

Usage

equilibrate(aout, balance=NULL, loga.balance=NULL,
    ispecies=1:length(aout$values), normalize=FALSE, as.residue=FALSE,
    method=c("boltzmann", "reaction"))
  equil.boltzmann(Astar, n.balance, loga.balance)
  equil.reaction(Astar, n.balance, loga.balance)

Arguments

aout

list, output from affinity

balance

character or numeric, how to balance the transformations

ispecies

numeric, which species to include

normalize

logical, normalize the molar formulas of species by the balancing coefficients?

as.residue

logical, report results for the normalized formulas?

Astar

numeric, affinities of formation reactions excluding species contribution

n.balance

numeric, number of moles of conserved component in the formation reactions of the species of interest

loga.balance

numeric, logarithm of total activity of balanced quantity

method

character, equilibration method to use

Value

equil.reaction and 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.balance, Astar, and loga.equil (the calculated equilibrium activities of the species).

Algorithms

The input values to equil.reaction and 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 (loga.balance). 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.

In 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 uniroot. 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.

In 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. If equil.boltzmann is called with balance other than 1, it stops with an error.

Details

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.reaction or 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:

• NULL: autoselect

• 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). If 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 normalize or as.residue is TRUE), otherwise equil.reaction is called. The default behavior can be overriden by specifying either boltzmann or reaction in method. Using 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.

References

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

See Also

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.reaction and equil.boltzmann to parallelize intensive parts of the calculations.

Examples

# NOT RUN {
## 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))$values
# the logarithms of activity for a total activity
# of the balanced quantity (C or CO2) equal to 0.001
loga.boltz <- equil.boltzmann(Astar, c(1, 1, 1), 0.001)
# calculated another way
loga.react <- equil.reaction(Astar, c(1, 1, 1), 0.001)
# probably close enough for most purposes
stopifnot(all.equal(loga.boltz, loga.react))
# the first ionization constant (pKa)
ipKa <- which.min(abs(loga.boltz[[1]] - loga.boltz[[2]]))
pKa.equil <- seq(0, 14, length.out=res)[ipKa]
# calculate logK directly
logK <- subcrt(c("CO2","H2O","HCO3-","H+"), c(-1, -1, 1, 1), T=25)$out$logK
# we could decrease tolerance here by increasing res
stopifnot(all.equal(pKa.equil, -logK, tolerance=1e-2))
# }