Global analysis of kinetic and equilibrium data produces better results. Often, global analysis makes the difference between correct and incorrect conclusions.
Fueled by the invention of a scanning spectrophotometer capable of producing 1,000 spectral scans per second and admonished by the pioneering kineticist, Professor Quentin H. Gibson to “pity the poor scientist who must deal with all the data your machine will vomit at him”, Drs. Richard J. DeSa and Iain B.C. Matheson developed one method after the other to resolve the thousands of millisecond snapshots into a meaningful set of starting, final, and intermediate species.
The near decade long development of this program has culminated in an instantaneously fast and extremely easy to use software package which allows the experimentalist to fit kinetic and equilibrium 3D, allowing correct conclusions to be made from multidimensional data in seconds. Below is a description of some of the math that we use in our calculations.
Factor analysis is a multivariate statistical method for extracting the information content from a data set. In the Olis case we have a large matrix consisting of absorbance etc. as function of both wavelength and time. Factor analysis, in the Olis case singular value decomposition (SVD), reduces this to 3 smaller matrices that contain all the information present in the data. The kinetic information is contained in a kinetic eigenvector matrix. Experience has shown that only a small number of eigenvectors are needed to describe the data. Olis uses 6. The various rate processes are stored over a small number of eigenvectors and a kinetic model may be fitted to those to extract the rate constants. Similarly the spectral information is stored in a set of 6 eigenvectors. Finally a vector of singular values is produced. Typically these start off at a high value and fall off rapidly to a value near zero. The number of singular values above zero is an indication of the number of species significantly present and is useful for model selection.
A representation of the original data may be obtained by a suitable matrix multiplication of the 2 eigenvector sets of vectors and the singular values. This reconstruction is the original less most of the noise. Thus the processes of SVD and reconstruction smooth the data. The 3 matrices are commonly used to store the information in the data in a much more compact form than the original; about 30 times smaller. It is also much easier to do the global kinetic fit on the compact kinetic set than the original large data set. Thus factor analysis cleans up the data, affords more compact storage, and greatly reduces the scale of the global fitting problem.
SVD is one of several methods for factor analysis and is generally recognized as being the mathematically most robust. The Olis modifications to SVD in no way modify the way in which it is used or the conclusions drawn from it. The only difference is a large gain in speed.
The Simplex Method is a linear fitting procedure to mathematical functions which may be applied to non-linear problems. It uses linear adjustment of the parameters until some convergence criterion is met. The term ‘simplex’ arises because the feasible solutions for the parameters may be represented by a polytope figure called a “simplex.” The simplex for the case of a function of N rates is stored as an (N+1)xN rectangular array. Each column of the array contains N rates and represents a vertex of the polytope. In the 2 rate case the simplex is a triangle, 3 rates a tetrahedron, and 4 or more rates a not easily envisaged multidimensional polytope. The algorithm is given an initial set of rates in the simplex array and proceeds to find the function minimum by a process of reflection expansion and contraction of the simplex. The algorithm invariably converges to a minimum following a series of contractions so that the final simplex contains very similar rates in each column and each row has a sum of squares which is the same to at least 4 significant figures.
The Simplex method differs from the well-known and widely used Levenberg-Marquardt and Gauss-Newton methods in that it does not use derivatives, which confers safer convergence properties to the Simplex method since it is much less prone to finding false minima. One of the more remarkable features of the Downhill Simplex algorithm is that no divisions are required. Thus the pesky “div/0 runtime error” is bypassed.
Algorithms for the exponentiation of a matrix have become available in recent years. It is not generally known that matrix exponentiation may be used to solve sets of kinetic differential equations where all rates are first order. Most kinetic processes of interest consist of a collection of first and pseudo-first order processes. Consider the simple first order decay:
dA/dt = -rate(1)*A = -kA.
The solution to this is A = Ao.e(-kt), i.e., the solution is of exponential form.
Consider now the case of A → B, for which the differential equations are:
dA/dt = -kA
dB/dt = kA,
which can be written in matrix form as:
A | B | |
dA/dt | -k | 0 |
db/dt | +k | 0 |
Here, the matrix columns represent the respective initial concentrations and the rows their derivatives with respect to time. Note that in this case the initial concentration of B is zero, and that the B column entries are zero because there is no back reaction.
In general such a matrix contains only rate constants and zeros. The available algorithms for calculation of the exponential of a matrix produce a solution which, when applied to the above matrix, corresponds to:
exp(-kt) | 0 |
1-exp(-kt) | 1 |
This matrix resulting from the exponentiation process requires further simplification. Recalling that exponentials are normalized to unity the initial concentrations of A and B are the 1 and 0. Accordingly the resultant matrix is multiplied by a vector with contents of 1 and 0 yielding the concentrations for Aand B at time t as exp(-kt) and 1-exp(-kt). This result is the same is obtained by the usual algebraic methods and for a case of N time points yields an Nx2 concentration matrix. This concentration matrix is then inverted or used to calculate the function as required.
Thus the differential equations have been solved by applying a special type of exponentiation to a matrix of rate constants representing the differential equations. Matrix Exponentiation is thus a more general kind of exponentiation: the beauty of the Exponential Matrix method is that it may be easily extended to much more complicated cases with many first order processes including reversible steps. Application to complex cases in practice is not computationally limited but is strongly limited in terms of signal to noise ratio of the data. In other word, increasing the number of significantly determined rates requires an increasingly higher signal to noise ratio.
The Exponential Matrix method is markedly superior to the numerical integration method of Runge-Kutta for the computation of complex arrays of first order processes. It may be readily shown for the simple and common case of A → B → C the exponential matrix method is more than 20 times faster than Runge-Kutta. Thus not only does Runge-Kutta have well known stiffness problems it is so much slower that it makes the program no longer real time.
Formation of Differential Matrix
Mechanism
k_{1} | k_{2} | |||
A | → | B | → | C |
Rate equation in differential form
dA/dt = -k_{1}(A) | +0 | +0 |
dB/dt = +k_{1}(A) | -k_{2}(B) | +0 |
dC/dt = 0 | +k_{2}(B) | +0 |
Matrix derived from above by removing the concentration terms
-k1 | 0 | 0 |
+k1 | -k2 | 0 |
0 | +k2 | 0 |
It is this matrix itself which is solved by Matrix Exponentiation!
In this way, Matrix Exponentiation provides an exact solution of the rate equation.
We could say “instantaneous” and be fully accurate!
Today’s Olis GlobalWorks is the combined effort of Olis, Inc. and Evolution Software Design, Inc. to create the best 3D data handling and fitting software package in the world. The software includes an ever expanding selection of supported kinetic and equilibrium models, with nearly 50 are incorporated already, and a linkable module for acquiring data from a host of spectrophotometric equipment, including, but not limited to the Olis RSM 1000, Olis DSM CD Spectrophotometers, Olis Upgraded Spectrometers, and others.
- A→B Single Colored Species Cases
- A→B, Multiple Rates, Special Cases
- Two Species Cases
- Three Species Cases
- Four Species Cases
- Five Species Cases
Pure First Order Process | A→B |
Pure Second Order Process | A→B |
First Order Rise | A→B |
Polynomial Fit | A→B |
Pure Zeroth Order Process | A→B |
2 Exponentials, 1 Species | A→B |
2 Exponentials with Background, 1 Species | A→B |
3 Exponentials, 1 species | A→B |
3 Exponentials with Background, 1 Species | A→B |
First Order with Growing in | A→B |
Reversible First Order | A↔B |
First Order with Background | A→C, B |
Two First Order Decays | A→C, B→C |
Second Order with Growing in | A+A®B |
Second Order Decay with Background | A+A→C, B |
Normal Rise Fall | A→B→C |
Special Rise Fall | A→B+C, B→C |
Rise-Fall, Step 1 Reversible | A↔B→C |
Rise-Fall, Step 2 Reversible | A→B↔C |
Reversible Rise-Fall | A↔B↔C |
1 Exponential Growing In, 1 Independent Exponential Decaying | A→B, C→D |
1 Exponential Decaying, 1 Independent Exponential Growing in | A→D, B→C |
2 Independent Exponentials, Same Product | A→B, C→B |
2 Independent Exponentials with Background | A→D, B→D, C→D |
3 Independent Exponentials | A→D, B→D, C→D |
Irreversible Heterogeneous Second, 3 Concentrations | A+B→C |
Reversible Heterogeneous Second, 3 Concentrations | A+B↔C |
1 Exponential Rise, Second Order Fall | A→B, B+B→C+A |
Mixed Reduction Reactions | A→B, C→D, B+C→A+D |
Irreversible 4 Species Sequential | A→B→C→D |
4 Species Sequential with First Process Reversible | A↔B→C→D |
Reversible 4 Species Sequential | A↔B↔C↔D |
3 Independent Exponentials with Background | A→E, B→E, C→E, D |
4 Independent Exponentials | A→E, B→E, C→E, D→E |
4 Species Branching Reaction | A↔B, B↔C, B↔D |
Enzyme Reaction with Second Order Terms, 4 Species | A+B↔C↔D+A |
Irreversible 5 Species Sequential | A→ B→ C→ D→ E |
Reversible 5 Species Sequential | A↔ B↔ C↔D↔E |
4 Independent Exponentials with Background | A→ F, B→F, C→F, D→F, E→F |
5 Independent Exponentials | A→F, B→F, C→F, D→F, E→F |
Enzyme Reactions with Second Order Terms, 6 Species |
A+B↔C↔D↔E+A |
Please enter your name and email to begin downloading.
OLIS Globalworks is available as a free 30 day trial.