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We show how transformation group ideas can be naturally used to generate efficient algorithms for scientific computations. The general approach is illustrated on the example of determining, from the experimental data, the dissociation constants related to multiple binding sites. We also explain how the general transformation group approach is related to the standard (backpropagation) neural networks; this relation justifies the potential universal applicability of the group-related approach.

In this paper, on an important example of determining the dissociation constants related to multiple binding sites, we show that symmetries and groups can be useful in chemical computations.

In many practical situations, physical systems have _{6}H_{6} does not change if we rotate it 60°: this rotation simply replaces one carbon atom by another one. The knowledge of such geometric symmetries helps in chemical computations; see, e.g., [

Since symmetries are useful, once we know one symmetry, it is desirable to know all the symmetries of a given physical system. In other words, once we list the properties which are preserved under the original symmetry transformation, it is desirable to find

If a transformation

Similarly, if a transformation ^{−1} also does not change these properties. So, the set of all transformations that preserve given properties is closed under composition and inverse; such a set is called a

In this paper, we argue that symmetries can be used in scientific computations beyond geometric symmetries. To explain our idea, let us briefly recall the need for scientific computations.

One of the main objectives of science is to be able to predict future behavior of physical systems. To be able to make these predictions, we must find all possible dependencies _{1},…, _{n}_{1}, …, _{n}_{1}, … , _{m}_{1}, …, _{m}_{1}, … , _{m}_{i}

In general, to be able to predict the value of a desired quantity _{1}, … , _{n}_{1}, … , _{m}

first, we use the known observations
^{(k)} of _{i}_{i}

after that, we measure the current values _{i}_{i}_{1}, …, _{n}_{1}, …,_{m}

In scientific computation, the first problem is known as the

the forward problem is reasonably straightforward: it consists of applying a previously known algorithm, while

an inverse problem is much more complex since it requires that we solve a system of equations, and for this solution, no specific algorithm is given.

We assume that we know the form of the dependence _{1}, … , _{n}_{1}, …, _{m}_{i}_{1}, … , _{m}_{i}

In the idealized case when we can ignore the measurement uncertainty, the measured values
^{(k)} coincide with the actual values of the corresponding quantities. Thus, based on each measurement _{1}, … , _{m}

In general, we need _{1}, … , _{m}_{1}, … , _{m}

The dependence _{1}, … , _{n}_{1}, … , _{m}_{i}

Once the measurements of the quantities

we have a transformation ^{m}^{m}_{1},…, _{m}_{1},…, _{m}

we know the measured values _{meas} = (^{(1)}, … , ^{(m)});

we want to find the tuple _{meas}.

One way to solve this system is to find the inverse transformation ^{−1}, and then to apply this inverse transformation to the tuple _{meas} consisting of the measured values of the quantity ^{−1}(_{meas}).

So far, we have considered the ideal case, when the measurement errors are so small that they can be safely ignored. In most practical situations, measurement errors must be taken into account. Because of the measurement errors, the measurements results ^{(k)} and
^{(k)} and
^{(k)} = ^{(k)} + Δ_{k}

The formula
_{i}^{(k)} and
^{(k)} and
^{(k)} = y^{(k)} − Δ_{k}

Usually, the measurement errors Δ_{k}_{ki}

In many practical situations, measurement errors Δ_{k}_{ki}_{k}_{i}

In the general case, when the probability distributions of measurement errors may be different from normal, the Maximum Likelihood method may lead to the minimization of a different functional _{i}

In the absence of measurement errors, the measurement results coincide with the actual values, and thus, the solution _{i}_{i}^{(k)} and
^{(k)} and
_{i}_{i}_{i}_{i}_{i}

Thus, once we know how to efficiently solve the inverse problem in the idealized no-noise case, we can also efficiently extend the corresponding algorithm to the general noisy case:

first, we solve the non-noise system

then, we find the differences Δ_{i}

finally, we compute

In other words, the main computational complexity of solving the inverse problem occurs already in the non-noise case: once this case is solved, the general solution is straightforward. Because of this fact, in this paper, we concentrate on solving the no-noise problem—keeping in mind that the above linearization procedure enables us to readily extend the no-noise solution to the general case.

In many practical situations, we can make computations easier if, instead of directly solving a complex inverse problem, we represent it as a sequence of easier-to-solve problems.

For example, everyone knows how to solve a quadratic equation ^{2} + ^{4} + ^{2} + ^{4} + ^{2} + ^{2} + ^{2}. Then:

first, we solve the equation ^{2} +

next, we solve an equation ^{2} =

In general, if we represent a transformation _{1} ∘ … ∘ _{n}_{i}^{−1} can be represented as
_{i}

In transformation terms, solving an inverse problem means finding the inverse transformation, and simplification of this process means using compositions—and a possibility to invert each of the composed transformations. For this idea to work, the corresponding class of transformations should be closed under composition and inverse,

In a transformation group, the multiplication of two transformations ^{−1}.

The inverse problem of scientific computations—the problem of estimating the parameters of the model which are the best fit for the data—is often computationally difficult to solve. From the mathematical viewpoint, this problem can be reduced to finding the inverse ^{−1} to a given transformation. The computation of this inverse can be simplified if we represent _{1} ∘ … ∘ _{N}^{−1} as

An inverse problem of interval computations consists of finding an inverse ^{−1} to a given transformation ^{−1}, we try to represent _{i}

Which transformations are easier to invert? Inverting a transformation ^{m}^{m}_{k}(_{1}, … , _{m}^{(k)} with _{1}, … , _{m}

The simplest case is when we have a system of linear equations. In this case, there are well-known feasible algorithms for solving this system (

For nonlinear systems, in general, the fewer unknowns we have, the easier it is to solve the system. Thus, the easiest-to-solve system of non-linear equations is the system consisting of a single nonlinear equation with one unknown.

We would like to represent an arbitrary transformation

We are interested in transformations

(reversible) linear transformation and

transformations of the type (_{1}, … ,_{n}_{1}(_{1}), …, _{m}_{m}

One can easily check that such transformations form a group

To analyze which transformations can be approximated by compositions from this group, let us consider its closure
_{1},…, _{n}_{1}(_{1}), …, _{m}_{m}

By definition of the closure, this means that any differentiable transformation ^{m}^{m}

The same arguments show that we can still approximate a general transformation if, instead of _{i}_{i}

Linear and component-wise transformations are not only computationally convenient: from the physical viewpoint, they can be viewed as

Our objective is to find the tuple of the parameters _{1}_{m}_{meas}. Our idea is to find the inverse transformation ^{−1} and then to compute ^{1}(_{meas}).

Once we know how to represent the transformation _{1} ∘ … ∘ _{N}_{1}, … , _{N}^{−1}(_{meas}) as
_{meas} and by sequentially applying easy-to-compute transformations

For this method to be useful, we need to be able to represent a general non-linear transformation ^{m}^{m}

In some cases, the desired representation can be obtained analytically, by analyzing a specific expression for the transformation

To obtain such a representation in the general case, we can use the fact that the desired compositions

we start with the input layer, in which we input _{1}, … , _{m}

in the first processing layer, we apply the transformation _{N}_{N}

in the second processing layer, we apply the transformation _{n}_{−1} to the results _{n}_{N}_{−1}(_{N}

…

finally, at the last (_{1} to the results f_{2}(… (_{n}

A general linear transformation has the form
_{i}_{i}_{i}_{i}_{i}_{i}

This is a usual arrangement of neural networks. For example, in one of the most widely used 3-layer neural network with

we first compute

then, we apply, to each value _{k}_{0}(_{0}(_{k}_{0}(_{k}

finally, we compute a linear combination

(It is worth mentioning that a similar universal approximation result is known for neural networks: we can approximate an arbitrary continuous transformation (with any given accuracy) by an appropriate 3-layer neural network,

Neural networks are widely used in practice; one of the main reasons for their practical usefulness is that an efficient _{ki}_{i}

As we have mentioned, once such a representation is found, we can invert each of the components and thus, easily compute ^{−1}(_{meas}),

The

Let us show that such a simpler application is possible for a specific important problem of chemical computations: the problem of finding reaction parameters of multiple binding sites.

When there is a single binding site at which a ligand L can bind to a receptor R, the corresponding chemical kinetic equations L + R → LR and LR → L + R with intensities ^{+} and ^{−} lead to the following equilibrium equation for the corresponding concentrations [L], [R], and [LR]:

The presence of the bound ligands can be experimentally detected by the dimming of the fluorescence. The original intensity of the fluorescence is proportional to the original concentration [R]^{(0)} of the receptor; since some of the receptor molecules got bound, this original concentration is equal to [R]^{(0)} = [R] + [LR]. The dimming is proportional to the concentration [LR] of the bound receptor. Thus, the relative decrease in the fluorescence intensity is proportional to the ratio

Let us now consider the case of several (_{(}_{s}_{)}. In these terms, for example, the molecule in which two ligands are bound to the first and the third sites will be denoted by L_{(1)}L_{(3)}R. For each binding site _{(}_{s}_{)}R and L_{(}_{s}_{)}R → L + R with intensities
_{(}_{s}_{′)}R → L_{(}_{s}_{)}L_{(}_{s}_{′)}R and L_{(}_{s}_{)}L_{(}_{s}_{′)}R → L + L_{(}_{s}_{′)}R have the same intensities
_{−}_{s}_{+}_{s}_{+}_{s}_{−}_{s}_{−}_{s}_{+}_{s}

These summarized reactions lead to the following equilibrium equation for the corresponding concentrations [L], [R_{−}_{s}_{−}_{s}

Similarly to the case of the single binding site, the presence of bound ligands dims the fluorescence. Let _{s}_{s}_{s}_{s}_{+}_{s}_{s}

The original intensity of the fluorescence ^{(0)} of the receptor: ^{(0)}, where for every ^{(0)} = [R_{−}_{s}_{+}_{s}

The problem is to find the values _{s}_{ds}^{(k)} for different ligand concentrations [L] = ^{(k)}, and we want to find the values _{s}_{ds}

The system (2) is a difficult-to-solve system of nonlinear equations with 2

By adding all _{ds}^{S}_{S}^{S}_{S}_{−1} · ^{S}^{−1} + … + _{1} ·

The equations
^{(k)} · ^{(k)}) = ^{(k)}),

This is a system of linear equations with 2_{i}_{i}

Once we solve this linear system and find the values _{i}_{ds}_{ds}_{ds}_{ds}

Finally, once we find all the values _{ds}_{s}

Thus, the decomposition of the original difficult-to-invert transformation into a composition of easier-to-invert transformations (linear transformations and functions of one variable) leads to the following algorithm for computing the parameters of multiple binding sites.

We start with the values ^{(k)} of the bound proportion corresponding to different ligand concentrations ^{(k)}. Our objective is to find the parameters _{s}_{ds}

first, we solve the linear system (4) with 2_{i}_{i}

we then use the computed values _{i}_{ds}

we then substitute the resulting values _{ds}_{s}

Our numerical experiments confirmed the computational efficiency of the new algorithm.

Geometric symmetries has been effectively used to simply scientific computations, in particular, computations related to chemical problems. In this paper, we show that non-geometric “symmetries” (transformations) can also be very helpful in scientific computations. Specifically, we show that the

This work was supported in part by the National Science Foundation grants HRD-0734825 and HRD-1242122 (Cyber-ShARE Center of Excellence) and DUE-0926721, by Grants 1 T36 GM078000-01 and 1R43TR000173-01 from the National Institutes of Health, and by a grant N62909-12-1-7039 from the Office of Naval Research. The authors are thankful to Mahesh Narayan for his help, to Larry Ellzey and Ming-Ying Leung for their encouragement, and to the anonymous referees for valuable suggestions.