MODEL-BASED DATA COMPRESSION: FROM DATA COMPRESSION TO INFORMATION CONDENSATION

HOLGER WALLMEIER

Aventis Research & Technologies GmbH & Co KG, Core Technology Area Biomathematics, Industrial Park Hoechst, G 515 A, D-65926 Frankfurt am Main, Germany.
E-mail: wallmeier@CRT.hoechst.com

Received: 23^rd January 2001 / Published: 11^th May 2001

ABSTRACT

Support of industrial research and development activities by computing and information technologies today is coupled to huge amounts of data. Therefore, data management is a very crucial aspect of successful application of information technologies. Various strategies are used to handle the situation, each of which has its merits depending on the type of data, the context, and the usage.
Apart from the very straightforward approach to distribute data on appropriate storage media of sufficient volume, there are three different 'philosophies' of data compression.

1. Non-lossy data compression
2. Lossy data compression
3. Model-based data compression

Types 1 and 2 are probably the most widely used because they do not necessarily introduce a bias into the compressed data. There are a number of methods known today that are fully reversible, or at least reversible to a large extent.
This is different for model-based data compression. The idea is useful for data being produced by dynamic, deterministic systems. Important is the existence of a model with well-defined data scheme and data structure. These model features can be used to condense the corresponding original data. Two examples from industrial research are presented.
First example is the representation of computer simulations of molecular ensembles by correlation functions. The second example is the representation of microbiological studies on pathogenicity by kinetic constants. In both cases, the underlying model together with methods to generate compressed data representations allows efficient interpretation of simulations or experiments, respectively.
High levels of data condensation provide a variety of opportunities to link results from research and development to auxiliary information from many different sources. Thus, powerful infrastructures for decision support can be created.
 

INTRODUCTION

Production of Data

The typical scenario of data generation starts from a device or automaton producing data (production), which will be recorded using some representation characteristic for the data and, of course, characteristic for the production itself. Based on this representation, data will be processed to extract the related information. In addition, the data may be transferred into a repository for later use.
Whenever the original production is deterministic, a faithful model of the original data production can be found, at least in principle. In such cases there is a unique data scheme and, furthermore, a well-defined analytical data representation. Usually, such models are given by a system of differential equations, the solutions of which define the data scheme. The way in which these solutions are determined also defines options of data representation. Extraction of information is then straightforward (Figure 1).
 

The advantage of the correspondence between original production and model production is that data scheme and data representation of the model can be used to generate a condensed representation of the original data. The information behind model data can usually be represented by few parameters.
The way in which this works will be shown by two examples. The first example is the computer simulation of molecular structures to analyze the stability of biomolecular complexes. In the second example, it will be shown how microbiological experiments with pathogenic bacteria can be analyzed in a very efficient way.

Representation and Condensation of data

Extracting and condensing information from data means creating a specific representation of the information. Basically, there are two different approaches to representing information. On the one hand, information can be mapped using predefined descriptor sets, thus creating specific profiles. On the other hand, information can be mapped in terms of relationships of the given object to known objects, which results, at least, in a delimiting view of the information. Genealogic aspects can be taken into account quite easily on a class and instance basis.
Both approaches offer several ways to condense information. Descriptor sets and profiles, for example can be handled using statistical methods such as clustering and classification, which also suggest strategies of visualization familiar from statistics and data mining.
Representing information by specifying relationships is first of all a simple and direct way of classifying objects based on similarity. In addition, this concept directly leads into the world of semantic networks.
 

AN EXAMPLE FROM MOLECULAR MODELING

Molecular modeling can be very useful to assess questions regarding, very generally speaking, stability and affinity of molecular systems. A very powerful, even though 'expensive' tool is the simulation of the dynamics of molecular systems. The underlying paradigm is based on perturbation theory. Simulations can be considered as computer experiments that allow the study of the response of a given system (molecular model) to some defined perturbation. The perturbation applied most frequently is just the kinetic energy of the N particles associated with a given temperature T according to [1]

Such simulations show the time evolution of the given system under the thermodynamic conditions specified and allow us to judge the stability of the given structural alignment, constitution, or conformation relative to some reference state. For this reason, simulation of molecular dynamics is a quite popular way of performing conformation-searches, especially for large molecular systems. By extending the analysis to the various aspects of entropy, affinity can also be estimated, at least on a molecular level.

A Model for Dynamical Affinity of Molecular Systems

In practice, molecular dynamics simulations are performed by discretized integration of the respective equations of motion. [2] Since the characteristic frequencies of all relevant degrees of freedom must be resolved by the integration step-size, simulations of molecular dynamics are usually very lengthy and produce huge data sets (trajectories) if applied to large systems.
There is, however, a way to avoid very long simulations. The idea is based on the concept of collective modes of oscillation, which exist in stable molecular alignments. Indeed, the existence of such collective modes can be taken as a criterion for stability, because they make the difference between an unstable scattering state and a stable bound state of a molecular aggregate. According to quantum mechanics, their respective Eigenfrequency can identify such modes. Using a so-called Drude model, [3] which was originally developed for the electronic dispersion interaction of atoms and molecules by London, [4] this can be shown quite easily. Interacting molecules are represented by pairs of coupled harmonic oscillators (Figure 2).

For simplicity, we take a pair of one-dimensional, coupled, identical harmonic oscillators positioned on the z-axis. The corresponding Hamiltonian is given by

where m is the mass of each oscillator, K the force constant and a the coupling constant, which is a function of the distance between the equilibrium positions of the oscillators. After separation of variables one has

The first term represents the coherent motion of the center of gravity of the pair of oscillators and the second term the relative 'breathing' motion. Since both oscillators have a ground state frequency w₀, coupling results in a symmetric split of energy levels as shown in the following scheme (for a > 0), (for a < 0 in reversed order):

The energy of an oscillator is , so that the splitting is given by

For small |a| one has

which is a typical second order, resonance-like effect.
Coming back to classical mechanics, one can calculate the sum over states (|a| << w₀) of the system

The Helmholtz free energy is

and since

and

it is clear that

Therefore, in terms of thermodynamics, coupling of oscillators adds a contribution to the energy of the overall system, which is mainly an entropy effect. It should be noted that the difference of the energy levels is independent of the sign of the coupling constant, since it is proportional to a². The energetic order of w₊ and w_-, however, is a function of the sign of the coupling constant. By analogy to the analysis of the (electronic) dispersion interaction by London, [4] this contribution to the entropy of molecular complexes can be called mechanical dispersion or, because of its stabilizing effect, dynamical affinity.

Tracing Dynamical Affinity in Molecular Dynamics Simulations

In a molecular dynamics simulation dynamical affinity can be traced, mapping coherent and breathing motion by correlation functions.

For harmonic oscillators, one can define the autocorrelation functions for coherent and breathing motion

Now, it can be shown that the second derivative of these correlation functions is -w² for zero correlation time (d=0). This means that the whole simulation can be condensed to just two independent numbers, w₊, w_-, and perhaps Dw=w₊-w_-.
G⁺ and G^- are determined by selecting two centers (atoms or groups of atoms) i and j. The only condition to meet, is that i and j should be influenced in their dynamics by both, the coherent and the breathing mode.
Since w₊ and w_- are determined from the trajectories of the full simulation ensemble, they are frequencies from the phonon spectrum of the whole system and not just frequencies of local molecular vibrations. In fact, splitting into w₊ and w_- is a sensitive indication of the existence of a common, non-local mode of vibration for both oscillators. This of course shows that the interaction between the molecules has lead to a stable bound state and not an unstable scattering state.

Streptavidin and Biotin

The example given below is a complex of two biomolecules, the protein Streptavidin and the vitamin Biotin. They form a specific complex with the largest binding constant known between biomolecules in nature. Therefore, this system is frequently used for immobilization of biomolecules. Surprisingly, experimental studies with molecules slightly different from Biotin show significant loss of stability and document the high specificity of the Biotin/Streptavidin complex.
 

For example, 2-(4'-hydroxyphenylazo)-benzoic acid (HABA) also binds to streptavidin, but with a binding constant which is 9 (!) orders of magnitude lower.

The thermochemical data measured for these complexes are given in Table 1. [5]
 

Apart from the remarkable values of the binding constants, it should be noted that the sign of the entropy contribution to the free binding energy changes going from Biotin to HABA. This is an indication of a change in the role of entropy.
From the theoretical point of view, it is of course a challenge to model such a system. Fortunately, crystal structures of both complexes have been published. [6] Molecular dynamics simulations starting from these crystal structures have been run using the AMBER 3.0 [7] force field in NVT ensembles with water and counterions at 300 K. The ensembles have been thermalized during 30 psec simulations. Subsequently, another 15 psec were used to sample the trajectories from which oscillator correlation functions have been estimated. Table 2 summarizes the results of the simulations.
 

For Biotin, four different orientations of the ligand in the binding pocket of Streptavidin have been simulated, for two HABA (Table 3). Columns 2 and 3 show the frequencies of the coupled oscillator motions derived from the autocorrelation functions G⁺ and G^-. For the crystal structure orientation of Biotin (1) the coherent motion has the lower frequency and the breathing motion is significantly faster.
 

In the first row of Table 2 the Biotin-results of a simulation without water and counterions are given. The values do not differ very much from the results for the solvated system, which indicates the robustness of the method.
The interesting result is that the entropy contribution from oscillator coupling found in the molecular dynamics simulations shows the same relationship between Biotin and HABA as does the experimentally determined quantity T×DS. The agreement is 13% with respect to the experimental value, which is adequate for the force field chosen, the size of the simulation ensembles, and the simulation time.
 

This underlines the role of oscillator coupling as indicator for stability of a given molecular alignment. At the same time it demonstrates the potential of data reduction that is given by this approach.
In terms of model-based data compression, we have the following situation. The original data production is the molecular dynamics algorithm in combination with the force field model of the system. The trajectories are the original data. Now, the model production is given by the coupled oscillators, the corresponding data scheme by the oscillator correlation functions, and the data representation by the oscillator frequencies. The representation of the information, i.e. the descriptor of stability, is given by the level splitting, calculated from the frequencies.
 

AN EXAMPLE FROM BIOMETRY

Quite a different approach to model based data compression is possible in the area of kinetic studies of bacterial pathogenicity. Such studies are very important in infectious disease research. In a very general view, the key issue is the interaction between pathogens and the hosts they infect. Besides the medicinal aspects of infection, pathogen-host interactions are the primary focus of target and lead compound search in the pharmaceutical industry. It is a complex phenomenon with several degrees of freedom.

Dynamics of Infectious Disease Progression

Progression of an infectious disease is, in a generalized sense, always the result of several types of growth processes, which are characteristic for different phases of disease progression. [10] If one wants to identify targets for anti-infective drugs, the early phases of disease progression are of special interest.
The first phase is the invasion of the pathogen. This is some kind of transport phenomenon, which often is coupled to specific surface interactions and recognition steps.
What follows is a phase of establishment that usually results in a growth of the pathogen population. In this phase chemical communication between pathogen and host may occur, which can facilitate the pathogen's establishment significantly. The chemical 'messages' pathogens send to the host are called virulence factors. Typically, they serve to subvert normal functions of the host cells. Sometimes they have an immuno-suppressive effect. [11]
Next is the formation or enrichment of so-called pathogenicity factors. Very often they are toxins secreted by the pathogen. But also bacterial enzymes, which, for example cause necrotic degradation of host tissue belong to this class of factors.
Last but not least, the development of disease symptoms is related to the amount of pathogenicity factors formed. In all theses phases, however, there is some kind of host response to defend against the pathogen. For more complex host organisms it is an immune response.
 

The scenario described above can be summarized in terms of the categories pathogenicity, virulence, and susceptibilty. Even though in literature pathogenicity and virulence are often used synonymously, a distinction based on genomical and disease progression considerations is possible. Pathogenicity is first of all a property of a pathogen that manifests in the formation of pathogenic factors like, for example toxins. [12] This, of course, depends on genotypic, as well as phenotypic conditioning of the pathogen. To be specific, what matters is type and amount of pathogenicity factors produced by the pathogen inside, or in contact with the host. The amount of factors formed, however, also depends on the size of the pathogen population inside the host, which, in turn, depends on genotypic and phenotypic conditioning of the pathogen.
 

Due to host response, however, pathogen multiplication also depends on genotypic and phenotypic conditioning of the host. In principle, there are two degrees of freedom for the pathogen. These are, on the one hand its ability to produce pathogenicity factors, and on the other hand the size of population of pathogenicity factor producing pathogens inside the host.
Since virulence factors are often host specific, many authors refer the notion virulence to the combined effect of pathogenicity factor formation and population growth.
The third degree of freedom (see Figure 5) is the host's susceptibility to infection by the pathogen. Here, genotypic and phenotypic conditioning of the host are the important features.
 

Any research in the field of infectious diseases aimed at understanding the large variety of strategies pathogens have developed during evolution must analyze the kinetics related to the different phases. First of all, descriptors have to be identified that allow us to follow the individual processes by experimental measurements (see Table 4).
 

A key problem in handling living organisms is reproducibility. Usually, this is taken care of by running replicate experiments and forming averages. In addition, time-resolved measurements are necessary to analyze the associated kinetics. To do so, the following model assumptions are useful.

A Model for Infectious Disease Dynamics

The normal way to measure pathogenicity starts from a set of N₀ host organisms, which are infected. In the course of the experiment, decrease of the host population is measured. Typically, one obtains a sigmoid curve (Figure 6), which can be represented by the solutions of the following differential equation (DE). It is called the logistic, autocatalytic, or autokatakinetic differential equation [14]

describing growth processes with feedback. It is the equation of an exponential growth, which is modified by the second term in the square brackets. This second term depends on the population N at time t and constitutes the feedback. It can be agonistic (g<0), as well as antagonistic (g>0). The general form of the solution is

With the integration constant N₀, the initial size of the population, plus the rate constant k, and the feedback constant g there are three independent parameters. The combination of N₀>0 and a negative value of k describes the decrease of a population (Figure 6).
In contrast, vanishing N₀ together with a positive value of k describes a population that grows into a saturation state. With such parameters a growing pathogen population can be described (Figure 7).

Applying this equation to infection experiments, one has, first of all, equation (11) for the decrease of the host population. According to the considerations outlined above, it is easy to imagine that the kinetic constant k in fact depends on the growing pathogen population P(t) and is thus a pseudo-constant. Therefore,

The growth of the pathogen population may either be unrestricted (free exponential growth)

or restricted,

reaching a saturation level due to host response. As mentioned above, P₀ is small, and k is positive.
The simplest form of combining the two processes is to set

which, for example results in the situation shown in Figure 8.

It is obvious that the effect of the growing (not constant) pathogen population can be seen as a deformation of the host population curve. The degree of deformation increases with h. There is, however, a further type of deformation of the host population curve. It comes from the feedback term and can be seen in Figures 9 and 10. This certainly reflects host conditioning.
 

SUMMARY

Whenever it is possible, model-based data compression serves two purposes. First of all it can be a great help to condense even huge data sets to very few numbers. Furthermore, the definition of the model necessary for compression is a very challenging step that often helps to gain deeper insights into the matter. It can reveal inconsistencies and facilitate the recognition of unknown phenomena. Together with the condensed data it offers possibilities to represent the information behind the data in a very efficient way.
However, any kind of modeling is an abstraction and idealization. A model always has to skip part of the reality. This of course limits the applicability of model-based data compression and defines the due diligence that must be applied using it.
 

REFERENCES AND NOTES

[1] McQuarrie, D. A.;Statistical Mechanics, Harper & Row, New York, 1976.
[2] van Gunsteren, W. F.; Weiner, P. K. (Eds.), Computer Simulation of Biomolecular Systems, ESCOM, Leiden, 1989.
[3] Drude, P. K. L.; The Theory of Optics. Longman, London, 1933.
[4] London, F. Z. Physik 1930, 63, 245 ; Z. phys. Chem. B 1930, 11, 222; Trans. Faraday Soc. 1937, 33, 537.
[5] Weber, P. C.; Ohlendorf, D. H.; Wendoloski, J. J.; Salemme, F. R. Science 1992, 243, 85.
[6] Weber, P. C.; Wendoloski, J. J.; Pantoliano, M. W.; Salemme, F. R. J. Am. Chem. Soc. 1992, 114, 3197.
[7] Weiner, S. J.; Kollman, P.A.; Case, D. A.; Singh, U. C.; Ghio, C.; Alagona, G.; Profeta, S.; Weiner, P. J. Amer. Chem. Soc. 1984, 106, 765.
[8] Brookhaven PDB entry code
[9] 'in vacuo' (without water and counterions) simulation of the crystal structure of the Streptavidin/Biotin complex
[10] Finlay,B. B.; Falkow, S. Microbiol. Rev. 1989, 53, 210.
[11] Mescas, J.; Strauss, E. J. Emerg. Infect. Diseases 1996, 2, 271.
[12] Toxin is considered here to be not only a substance produced endogenically by the pathogen, but also toxic substances that are formed by host response to the pathogen.
[13] Measured as CFUs (colony forming units). See, e.g. Brock, T. D. Biology of Microorganisms, Prentice Hall, London 2000.
[14] Dost, F. H. Grundlagen der Pharmakokinetik, Georg Thieme Verlag, Stuttgart 1968; Nisbet, R. M.; Gurney, W. S. C. Modelling fluctuating populations, Wiley, New York, 1982; Skehan, P. Growth 1986, 50, 496; Renshaw, E. Modelling biological populations in space and time, Cambridge University Press, Cambridge, 1991; M. Marusic, M.; Bajzer, Z.; Vuk-Pavlovic, S.; Freyer, J. P. Bull. Mathem. Biology 1994, 56, 617.
[15] Mahajan-Miklos, S.; Tan, M. W.; Rahme, L. G.; Ausubel, F. M. Cell 1999, 96(1), 47; Mahajan-Miklos, S.; Rahme, L. G.; Ausubel, F. M. Mol. Microbiol. 2000, 37(5), 981.
[16] Dennis, J. E.; Schnabel, R. B. Numerical Methods for Unconstrained Optimization and Non-linear Equations, Prentice-Hall, 1983. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes in C, Cambridge University Press, Cambridge 1992.