Optimized derivation of transfer functions and a software giving it. Application to biological systems

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Abstract

Today, the transfer function between any two nodes of any linear flowgram are obtained either by applying the step-by-step classical elimination of nodes and branches or by applying the Mason’s rules, both of them being manual, tedious in many cases procedures and therefore prone to human errors. In this paper we facilitate the derivation of the transfer function in two different ways. The first one consists on reducing, when it were possible, the flowgram obtaining, manually, a simpler but equivalent subflowgram to which we apply the above mentioned procedures. Thus the time elapsed in the process as well as the probability of mistakes commission diminishes. The second way consists in deriving the transfer function using a user friendly software developed by us in this paper and that apply in an automatic form the Mason’s rules to the flowgram under study to derive any of the possible transfer function without errors and in a very short process time. In each case, the software works on the equivalent subflowgram obtained during the computer process in the same form as it is manually obtained. Finally, we apply the tool here presented to some examples of biological systems.

Introduction

During the last decades the application of methods of dynamic systems analysis and mathematical modelization are playing a growing important role concerning to different problems of the Science and Engineering. The techniques, of a great efficacy in all of cases, were fundamentally developed to their application to technological problems of information transfer, communication and control [1], [2], [3], [4], [5], [6]. They can also be applied to many of the most important functions occurring in processes in living organisms and too many of physical and chemical processes. It is not surprising, therefore, that the above mentioned techniques are increasingly applied to different fields of the Biology, Chemistry and Biochemistry [7], [8], [9], [10], [11], [12], [13], [14], [15], [16].

Any dynamic linear system may be represented by a signal-flow graph also merely called flowgram. A graph consists in a set of points, called nodes, which are connected by a set of directed and weighted edges or branches. One of the most important tasks in analysis of linear dynamic systems is the derivation of the so-called transmission or transfer function between any two nodes of the corresponding flowgram.

The method of signal-flow graphs to study dynamic linear systems was discovered by Mason [1] and has been widely used for the analysis of electrical and electronic circuits [1], [2], [3], [17] and also in the analysis of biological, medical and biochemical systems [7], [8], [9], [10], [18], [19], [20], [21], [22], [23]. But our main interest in this contribution is to obtain easy obtaining of the transfer functions from the corresponding flowgram for its application to biological systems. An important example of different biological fields in which the flowgrams and transfer functions obtained from them is the Metabolic Control Theory [8], [9], [24], [25], [26], which is based on the flow control coefficients and concentration control coefficients. The control coefficients are related to the elasticity coefficients by means of linear algebraic equations whose unknowns are the control coefficients. By solving these equations the control coefficients are calculated in terms of the elasticity coefficients. Other important fields of biological interest to which the flowgrams (signal-flows graphs) are applicable are all those ones which can be described by a system of linear differential equations, such as the kinetic behavior of linear enzyme systems, because the set of linear differential equations can be transformed, by using the Laplace transformation, in a set of linear algebraic equations susceptible to be solved which the flow-signal graph method [10].

A major advantage of the method of signal-flow graphs is that the corresponding linear system under study can be represented in a graphic manner directly from the configuration of the system. This also provides a visual framework for analyzing the cause–effect relationship of the individual components of the system and thus is aids in developing an intuitive understanding of it.

To each signal-flow graph or flowgram correspond different possible so-called transfer functions or flowgram transmissions. These transfer functions can be obtained from the flowgram either applying classical rules for step-by-step successive elimination of nodes and branches of the flowgram [1], [2], [3], [27], [28] or directly applying the well known Mason’s rules [1], [2], [3], [10] The step-by-step eliminations are a laborious and tedious task. The Mason’s rules make this task easier. Nevertheless, also the application of these rules, which necessarily must be performed manually, i.e. by inspection of the flowgram, can become, even for flowgrams of not too complexity, very laborious and, therefore, prone to human errors.

Therefore, the main objectives of this contribution are: (1) To facilitate the obtaining of the transfer function between any two nodes of the flowgram applying the Mason’s rules, not to the complete flowgram but only to that part of the flowgram having influence on the wanted transfer function, what moreover world be the advantage of living the expressions of the transfer function in the most simplified form. (2) To develop a software furnishing the transfer function between any two nodes of any flowgram using the systematic and automatic application of the Mason’s rules. (3) To apply the resulting software, as example, to several biological system.

Section snippets

Materials and methods

To reach from the set of linear differential equations describing a linear dynamic system to the corresponding algebraic set of algebraic linear equations and its associated flowgram, the Laplace transformation method was used. To implement the software called TRANSFU we have used normal programming abilities and in a computerized and automatic way to apply the Mason’s rules to a subflowgram equivalent to the original one, so that the, program furnishes the transfer function between any two

Flowgrams and transfer functions

To better reach the purposes of this contribution we firstly will slightly review in this section the main concepts and definitions of the method of flowgrams fundamentally based on contributions of Mason and Zimmermann [1], [2] and Puente [3].

Optimization of the acquisition of the transfer function between two nodes in a flowgram

The application of the Mason’s rules to obtain, by using Eq. (5), the transfer function, Te,f, between any two nodes xe and xf of a flowgram, works on the complete flowgram. One way to facilitate the task is to work only on that part of the flowgram whose nodes and branch transmissions have influence on the transfer function Te,f. This is the objective of this section where we must begin introducing an additional notation to that already used previously.

Software to apply the Mason’s rules

The Mason’s rules, in spite they supposed a great advance in the systematic reduction of flowgrams is a procedure that, applied manually, is slow and prone to human errors even for flowgrams with reduced complexity. But, fortunately, due to the sequential, recurrent and systematic character of these rules it is possible to develop, as we have in this contribution, a software giving directly the transfer function, Te,f, between any two nodes xe and xf of any flowgram. The implementation of this

Results and discussion

In Section 3 above we summarized, in an ordered way and oriented to our own contribution the more important concepts related with the flowgrams, the transfer functions, the manual step-by-step reduction of a flowgram and the Mason’s rules for obtain the transfer function of a flowgram. The present section is devoted to comment and discuss the results obtained in Sections 4 Optimization of the acquisition of the transfer function between two nodes in a flowgram, 5 Software to apply the Mason’s

Acknowledgement

This work was supported by grants from the Comisión Interministerial de Ciencia y Tecnología (MCyT, Spain), Project No. BQU2002-01960 and from Junta de Comunidades de Castilla-La Mancha, Project No. PAI-05-036.

References (34)

  • S.J. Mason et al.

    Electronics Circuits, Signals, and Systems

    (1960)
  • E.A. Puente, Regulación Automática, Universidad Politécnica de Madrid, Escuela Técnica Superior de Ingenieros...
  • J. Vlach

    Computer Methods for Circuit Analysis and Design

    (1994)
  • K. Ogata

    Ingeniería de control moderna

    (2003)
  • F. Matía et al.

    Teoría de sistemas

    (2005)
  • L. Finkelstein et al.

    Mathematical Modelling of Dynamic Biological Systems

    (1985)
  • A.K. Sen

    Metabolic control analysis. An application of signal flow graphs

    Biochem. J.

    (1990)
  • Cited by (0)

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