## Introduction

Differential growth in cylindrical organs, such as roots and hypocotyls, is a complex process that involves changes in transcription and dynamic alteration of protein expression patterns (Muday, 2001; Friml & Palme, 2002; Friml *et al*., 2002; Blilou *et al*., 2005; Teale *et al*., 2005). Quantitative analysis of growth and differential growth is a prerequisite to understand the molecular organization of this process. Several concepts have been used to characterize the curvature of cylindrical organs, the three most notable being the differential relative elemental growth rate (REGR) distributions (Silk & Erickson, 1978; Silk, 1989; Ishikawa & Evans, 1993; Zieschang *et al*., 1997; Mullen *et al*., 1998a), the curvature (κ) (Silk & Erickson, 1978; Selker & Sievers, 1987; Silk, 1989; Zieschang & Sievers, 1991) and distributions of the curvature angle (Mullen *et al*., 1998b; 2000; Wolverton *et al*., 2002a,b). Differential REGR distributions and the rate of change of curvature (dκ/d*t*) have been shown to be equivalent (Silk & Erickson, 1978; Silk, 1989; Zieschang *et al*., 1997). However, differential REGR profiles are prone to errors because the traditional method used to measure REGR distributions relies on a relatively small amount of markers, and interpolation schemes need to be applied (Peters & Bernstein, 1997). Moreover, the determination of curvature production through differential REGR profiles is critical, because the coordinates of the profiles have to be matched correctly (nontrivial, for example, for curved root geometry and easily resulting in artifacts). As will be shown in the ‘Description’ section, κ and dκ/d*t* are not suited to describe the change of orientation of an organ and do not quantify sufficiently the production of curvature. However, quantitative relations of curvature production at specific regions need to be established to elucidate differential growth and organ curvature in gravitropism, phototropism and hydrotropism (Blancaflor & Masson, 2003; Eapen *et al*., 2005; Esmon *et al*., 2005). Therefore, the concept of the curvature angle distribution (Mullen *et al*., 1998b; 2000; Wolverton *et al*., 2002a,b) was extended here to find a suitable measure of curvature production [differential growth curvature rate (DGCR)]. Theoretical calculations presented below, show, for curvature occurring in a plane, the relation of the DGCR to dκ/d*t* and to differential REGR profiles. This concept is extended to describe curvature and torsion processes in three dimensions, and, in addition, the DGCR is applied in a simple model of root gravitropism and used to simulate two different cases of curvature production in root gravitropism (one and two sites of production). The recent proposal of the existence of two motors in root gravitropism is tested therewith (Wolverton *et al*., 2002a). The simulations presented here show that a suitable measure of curvature production is essential to be able to separate two motors that are located so closely as proposed [in the distal elongation zone (DEZ) and in the central elongation zone (CEZ); Ishikawa & Evans, 1993; Wolverton *et al*., 2002a], and confirm the need for a suitable measure of curvature production (DGCR).