RECTANGULAR HYPERBOLA FITTED TO MUSCLE FORCE-VELOCITY DATA USING THREE- DIMENSIONAL REGRESSION ANALYSIS

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1 Experimental Physiology (1994), 79, Printed in Great Britain RECTANGULAR HYPERBOLA FITTED TO MUSCLE FORCE-VELOCITY DATA USING THREE- DIMENSIONAL REGRESSION ANALYSIS Department of Pharmacology, Universit:y of Lund, S Lund, Sweden (MANUSCRIPT RECEIVED 12 MAY 1993, ACCEPTED I OCTOBER 1993) SUMMARY A simple method is described for fitting a rectangular hyperbola to force-velocity data derived from active muscle. The approach is based on three-dimensional regression analysis of the variables force, velocity and (force x velocity). The usage of the method is illustrated by practical applications. A pocket calculator is sufficient for the mathematical treatment. INTRODUCTION The relation between active force (P) and speed of shortening (V) has attracted much interest in muscle physiology, since this relationship contains much useful information concerning the basic mechanisms of muscle contraction (e.g. Huxley, 1957; Edman, 1979; Woledge, Curtin & Homsher, 1985). As first demonstrated by Hill (1938) the force-velocity relation in skeletal muscle can be adequately described as part of a rectangular hyperbola. Later studies have confirmed the hyperbolic nature of the force-velocity relation in single skeletal muscle fibres (Edman, Mulieri & Scubon-Mulieri, 1976), isolated papillary muscles (Edman & Nilsson, 1972) and various kinds of smooth muscle (Hellstrand & Johansson, 1975). More recent observations in single muscle fibres would seem to make it clear that the force-velocity relation is indeed a composite of two hyperbolic functions that are located on either side of a break point near 78 % of maximum isometric force (Edman, 1988). Several methods have been used in the past for fitting a hyperbolic function to experimental force-velocity data. The approaches generally used include graphical fitting of Hill's (1938) hyperbolic equation (Katz, 1939) and computer routines which systematically vary the numerical values of the parameters in Hill's equation to provide the best fit using the least squares criterion (Edman et al. 1976). In this paper a convenient method is described for fitting Hill's hyperbolic equation to experimental data using conventional statistical techniques. The approach is based on regression analysis of the variables P, V and PV in the hyperbolic equation. METHODS Principles of curvefitting Hill's equation describes a rectangular hyperbola and can be written in the form: (P+a)(V+b) = C, (1) in which P is force, V velocity of shortening and a, b and C are constants with dimensions of force, velocity and power, respectively. For calculating the numerical values of the three constants eqn (1) is planimerized to: PV = -bp-av+(c-ab) (2)

2 236.~8 4_ Force (P) (arbitrary units) Fig. 1. Diagram illustrating six data pairs of force (P) and velocity of shortening (V) that fall along a rectangular hyperbola of the following equation: (P+ 1) (V+ 2) = 12. The equation was derived as described in the text. This equation is rewritten as: Z= BXX+By Y+A, (3) where X = P, Y = V and Z = PV. By means of three-dimensional regression Bx, By and A can be determined from the experimental data using the following approach. For the theoretical basis of this calculation, see Fisher (1958). The parameters of Hill's equation are calculated as: b = -Bx a =-By C= A+BXBy. In the regression analysis used there are three possibilities of deriving the parameters Bx, By and A in that the equation of the plane (eqn (3)) can be obtained by minimizing the squared deviations about either X, Y or Z. The three methods yield very similar results. In this laboratory, minimization is carried out for deviations about Z, i.e. for deviations about PV (power). Calculation procedure Figure 1 shows an example of fitting a rectangular hyperbola to force-velocity data. The following values of P, V are used: (0, 10); (1, 4); (2, 2); (3, 1); (4, 04) and (5, 0). The calculation is performed in six steps: (1) X, Y and Z are defined as: X= P, Y = V and Z = PV. (2) The following sums (S) are determined (n = 6): S(X) = 15; S(Y) = 17-4; S(Z) = 12-6; S(XZ) = 27-4; S(XY) = 12-6; S(YZ) = 27-64; S(X2) = 55; S(Y2) = 12116; S(Z2) = (3) The sums listed under (2), i.e. S(X), S( Y), S(Z) etc., are used as follows for calculating the sums of squares or products of deviations from the means, e.g. [XY] = X(X- N)(Y- Y): [XY] = S(XY)-(l/n) x S(X) x S(Y); [XZ] = S(XZ)-(l/n) x S(X) x S(Z); [YZ] = S(YZ)-(l/n) x S(Y) x S(Z); [X2] = S(X2)-(l/n) x [S(X)]2; [y2] = S(Y2)-(l/n) X [S(y)]2; [Z2] = S(Z2)-(l/n) x [S(Z)]2. The numerical values of these quantities in the example given in Fig. 1 are: [XY] = -30-9; [XZ] = -4 1; [YZ] = -8-9; [X2] = 17 5; [y2] = 70 7 and [Z2] = 17-1.

3 FORCE-VELOCITY CURVE 237 A B c0*6 069 ' 0 \ o 0 U Force (units) Force (units) Fig. 2. Force-velocity data derived from an isolated muscle fibre in ordinary Ringer solution (A) and in a Ringer solution made hypertonic by addition of sucrose (tonicity 44% above normal). Hill's (1938) hyperbolic equation was fitted to data points truncated at 80 % of measured tetanic force. The following numerical values of the parameters of the hyperbolic equation were derived for the two curves: A, a/ps* = b/ Vm.ax = 0-267; B, a/p* = b/ Vmax = 0-606, where Vmax is the calculated maximum velocity of shortening and P* the calculated isometric force. Temperature: A, 3 3 C; B, 1 6 'C. (4) The parameters BX, By and A are calculated as: B - [XZ] [ y2] [yz] [Xy] AB =-200 x [X2] [ y2] _ [X]2 By = [YZ] -Bx[XY] By = [ y2] A = (I/n) x [S(Z)-Bx x S(X)-By x S(Y)] A = 10. (5) The parameters of the hyperbolic equation, a, b and C, are determined: a = -By = 100; b = -Bx = 200 and C = A+BxBy = 120. (6) Finally the squared multiple correlation coefficient is calculated: r2 = 1 _{[Z2] - Bx[XZ]-By[ yz]}/[z2]. The numerical value of r in the example used is 1. APPLICATIONS AND DISCUSSION Figure 2 illustrates two practical applications of the method used for fitting a hyperbolic function to force-velocity data. The method is based on multiple regression analysis and so provides an optimum fit of the hyperbola to the experimental data. The force-velocity data shown in Fig. 2 were collected from two frog single muscle fibres that were immersed in ordinary Ringer solution (A) and in hypertonic medium (B), respectively. The

4 238 force-velocity relation exhibits a break point in the high-force range (close to 80 % of the measured isometric force, P0) as described in more detail previously (Edman et al. 1976; Edman, 1988). For the curve fitting in Fig. 2, only data points at forces lower than 0 80 P0 were considered. An equation describing the entire force-velocity relation, i.e. including both curvatures, has been presented before (Edman, 1988). The examples illustrated in Fig. 2 exhibit markedly different curvatures within the main portion of the force-velocity relation, i.e. at low and intermediate loads. As can be readily seen, however, the data can be fitted well by a hyperbolic function in both cases. In multiple regression analysis, minimization can be performed about any of the three different axes X, Y or Z. In the approach described under Methods, only deviations about Z (= PV) are considered. However, virtually the same results are obtained if minimization is instead performed about X (= P) or Y (= V). It is essential to point out that the variables X, Y and Z are not independent of one another. Furthermore, the three variables are not normally distributed. This implies that the calculated values of standard deviation and correlation coefficient are not strictly acceptable for statistical evaluation. The numerical value of the standard deviation (and the correlation coefficient) nevertheless provides a useful measure of the spread of the data points about the hyperbola. The technique described in the present paper has a clear advantage over the commonly used graphical methods described by Katz (1939) in that it does not involve the use of P0, the isometric force. This is pertinent in view of the fact that the high-force region of the force-velocity relation deviates from the main part of the force-velocity curve (see above). The method of Katz (1939), which is based on the use of PO, is therefore inapplicable to fitting a hyperbola to observations at low and intermediate loads. The present technique was used by Josephson & Edman (1988) and by Edman (1988). In each of those papers, it was mentioned that comparisons were made between this technique and a more elaborate iterative procedure (Edman et al. 1976) and that the results of the two methods agreed closely. The present approach has an advantage over the previous method in that it is simpler and more convenient to use. This study was supported by grants from the Swedish Medical Research Council (project 14X- 184 to K. A. P. E. and project 14X-8664 to B. W.). REFERENCES EDMAN, K. A. P. (1979). The velocity of unloaded shortening and its relation to sarcomere length and isometric force in vertebrate muscle fibres. Journal of Physiology 291, EDMAN, K. A. P. (1988). Double-hyperbolic force-velocity relation in frog muscle fibres. Journal of Physiology 404, EDMAN, K. A. P., MULIERI, L. A. & SCUBON-MULIERI, B. (1976). Non-hyperbolic force-velocity relationship in single muscle fibres. Acta Physiologica Scandinavica 98, EDMAN, K. A. P. & NILSSON, E. (1972). Relationships between force and velocity of shortening in rabbit papillary muscle. Acta Physiologica Scandinavica 85, FISHER, R. A. (1958). Statistical Methods for Research Workers, 13th edn. Oliver & Boyd, Edinburgh, London. HELLSTRAND, P. & JOHANSSON, B. (1975). The force-velocity relation in phasic contractions of venous smooth muscle. Acta Physiologica Scandinavica 93, HILL, A. V. (1938). The heat of shortening and the dynamic constants of muscle. Proceedings of the Royal Society B 126, HUXLEY, A. F. (1957). Muscle structure and theories of contraction. Progress in Biophysics and Biophysical Chemistry 7,

5 FORCE-VELOCITY CURVE 239 JOSEPHSON, R. K. & EDMAN, K. A. P. (1988). The consequences of fibre heterogeneity on the force-velocity relation of skeletal muscle. Acta Physiologica Scandinavica 132, KATZ, B. (1939). The relation between force and speed of muscular contraction. Journal ofphysiology 96, WOLEDGE, R. C., CURTIN, N. A. & HOMSHER, E. (1985). Energetic Aspects of Muscle Contraction. Academic Press, London.