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Kinematics of axial plant root growth
J. theor. Biol. (1995) 174, 109–117
Kinematics of Axial Plant Root Growth Y. F† L. B‡ †Department of Renewable Resources, University of Alberta, Edmonton, Alberta, Canada T6G 2E3 and ‡Department of Crop and Soil Science, Oregon State University, Corvallis, OR 97331, U.S.A. (Received on 19 July 1994, Accepted on 19 October 1994)
Axial growth of plant roots can be modeled as the deformation of a one-dimensional continuum. In this study we present a theory of root elongation that incorporates different roles of cell division in the meristem and axial growth in the elongation region in overall root extension. The meristem is idealized as being located at the root tip, from which material points are produced continuously by cell division. Elongation takes place after material points leave the meristem. Each material point along the root axis is associated with a unique time at which it was produced. The total rate of root growth is the sum of the rate of elongation of all cells present plus the contribution from the meristem. Under steady-state conditions, the rate of elongation of a particular segment of the root is a function of its age only. The theory enables us to compare spatial patterns of steady-state root elongation with distribution of cell sizes along the root axis. The theory is also used to characterize the effects of soil water stress and temperature on root growth. Soil water stress reduces both the rate of meristem production and the subsequent cell elongation, resulting in a shorter elongation region. The effect of temperature is characterized by a temperature–time equivalence effect, leading to an elongation region with its length independent of temperature.
between material and spatial description of growth and the choice of appropriate coordinate systems. This theory has been applied in the analysis of root elongation (Gandar, 1983a, b; McCoy & Boersma, 1984; Sharp et al., 1988; Pahlavanian & Silk, 1988; Silk et al., 1989; Bret-Harte & Silk, 1994). The continuum theory, when applied to plant growth, is concerned essentially with the macroscopic properties of the plant tissue. In describing these properties, we may disregard the fact that a plant tissue must be regarded as discrete on the cellular scale. In attempting to explain these properties, however, it may be necessary to take into account the cellular structure of the plant tissue. Therefore, it is desirable that the material points of continuum theory correspond to particular groups of cells throughout the process of growth, so that comparisons between microscopic scale observations—such as rate of cell elongation and distribution of cell sizes along the root axis—and macroscopic scale observations—such as the rate of deformation—can be made (Silk et al., 1989).
Introduction Axial elongation of plant roots is the result of division and elongation of individual cells. Cell division occurs primarily in the meristem region near the root tip. Cell elongation occurs in an elongation region immediately behind the meristem (Green, 1976). The meristem of the primary root of corn (Zea mays) occupies a region of 1–2 mm long at the root tip. The elongation region extends over a length of about 10 mm from the root meristem (Erickson & Sax, 1956a, b; Sharp et al., 1988; Pahlavanian & Silk, 1988). Silk & Erickson (1979) examined the concept of plant tissue as a mathematical continuum and concluded that for processes involving gross morphological changes, such as cell expansion and division, the continuum assumption is appropriate, although the continuum assumption may not apply for those processes involving individual cell differentiation. They adopted the kinematic theory of conventional continuum mechanics to describe the growth of a plant tissue, and discussed the importance of the distinction 0022–5193/95/090109+09 $08.00/0
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The material points in a continuum are identified by their spatial position in the initial, undeformed, state. The deformation of the continuum is described as the motion of these material points through space. The assumption of non-penetrability of matter states that at any given moment each material point occupies a unique point in space and no two material points may occupy the same spatial position at the same time (Gandar, 1983a). The continuity assumption requires that the neighboring material points remain as neighbors during deformation. Silk & Erickson (1979) and Gandar (1983a, b) presented detailed discussions of these aspects of the continuum mechanics theory and their applications in plant growth studies. These studies established the foundation for a theory which has provided a framework for the quantitative description of growth of plant tissues. However, some fundamental aspects of plant growth remain to be addressed. A central concept of the classical continuum theory is the concept of initial, undeformed state. In a growing root, each cell may be at a different stage of development and deformation. At any given moment, there are cells at the whole spectrum of states ranging from those that have just finished the cell division cycle in the meristem to those that have reached maturity and stopped elongating. Therefore, the concept of the initial, undeformed, state may be applied to individual cells, or material points, as the state they were in at the time they entered the elongation region. It cannot be defined appropriately for a plant root as a whole. The second assumption central to the continuum theory is the equivalence of material and spatial points (Gandar, 1983a). The applicability of this assumption in plant root growth may also be questioned. The spatial position of a particular material point, or a group of cells, could only be defined unambiguously after it has left the meristem. If we regard the material points as having the same spatial position as the meristem before they leave the meristem, i.e. at the root tip, the assumption of material and spatial point equivalence fails since the spatial positions of the material points will not be unique. One may argue that the material points in the continuum theory may not correspond to the cellular structure of the plant root (McCoy & Boersma, 1984). They could be represented by the identifiable points on the surface of the root, such as those identified in streak photography (Erickson & Sax, 1956a; Sharp et al., 1988). This, however, would make it difficult to examine the continuum theory against the results of microscopic observations of cell division and elongation because a material point may represent different groups of cells
at different times. A second consequence of this lack of correspondence between the material points and the cellular structure would be the possibility of infinite elongation of a given material segment. To illustrate this point, examine the root segment between a mark on the tip of the root and a mark at a finite distance from the root tip, such as the base of the root. The elongation of this segment is regarded as a continuous deformation process. Since a root is capable of continued growth, a material segment so defined must be capable of infinite deformation, contradicting the fact that plant cells only grow to a finite size. These observations suggest that the assumption of equivalence between material and spatial points must be modified for plant roots in order to account for the effects of cell division in the meristem and the fact that cells grow to finite sizes. This study was initiated to find a set of simple kinematic equations for plant root growth by modifying the kinematic theory of conventional continuum mechanics. The approach used was to describe the axial growth of a plant root as the result of meristem cell division and subsequent cell elongation. The fact that the axial root growth represents a finite deformation of the plant root as a mathematical continuum is reflected in the resulting equations. In addition, the material points defined in this approach correspond closely to the underlying cellular structures of plant roots, allowing us to compare the results of the continuum theory to those of plant cell growth studies.
Theory In the discussions that follow, we adopt the convention that bold face letters represent vector quantities. Corresponding scalar quantities are denoted by italic letters. Capital letters will be used to represent quantities in material coordinates and corresponding quantities in spatial coordinates will be represented by lower case letters. In the continuum theory, the motion of a typical material point, denoted by X, is expressed as x=x(X, t),
te0,
(1)
where the x on the left-hand side is the spatial position of the material point X at time t; and the x on the right-hand side refers to a function with independent variables X and t. It is assumed that a unique inverse of (1) exists, which is expressed as X=X(x, t),
te0.
(2)
Equation (1) describes the spatial position of a typical material point as a function of time, and eqn (2) describes material points occupying a specific spatial location at different instances in time. Equations (1) and (2) describe the motion of those material points through space during deformation. The assumption of the equivalence of material and spatial points (Gandar, 1983a) is satisfied by the unique inverse relation between eqns (1) and (2). The continuity assumption requires that the neighboring material particles remain being neighbors during deformation. The equivalent mathematical statements of this assumption are lim [x(X+dX, t)−x(X, t)]=0, te0,
(3)
lim [X(x+dx, t)−X(x, t)]=0, te0.
(4)
dX 4 0
and dx 4 0
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where t identifies the unique time the material point X was produced by the meristem. The condition of non-penetrability of matter is stated for a plant root as: at any moment, tet, an arbitrary material point X(t) occupies a unique spatial position, and no two material points X(t1 ) and X(t2 ), t1$t2 , may occupy the same spatial position at times temax(t1 , t2 ). These statements are equivalent to stating that a unique inverse of (5) exists so that X(t)=X(x, t), tEt,
(6)
The continuity condition, as stated in eqns (3) and (4) for a general continuum, are similarly modified. For axial plant root growth, the continuity condition requires that lim [x[X(t), t]−x[X(t−dt), t]]=0,
dt 4 0
tEt,
dtq0,
(7)
tEt,
dxq0.
(8)
Silk & Erickson (1979) and Gandar (1983a, b) have presented detailed discussions of these aspects of the continuum theory and their applications in plant growth studies.
and
A conceptual process of root growth may be stated as follows. The growth region of a plant root consists of a meristem located at the root tip and an elongation region next to the meristem. Material points, or cells, are produced in the meristem by cell division. Cell elongation occurs after a cell leaves the meristem and continues until it reaches maturity. Both the spatial position and the relative distance between any two material points vary as growth progresses. It is necessary to describe completely the growth pattern of the root in order to identify material points by their spatial position or their position relative to other material points, such as the root tip (Gandar, 1983a). We recognize that the time at which a material point left the meristem increases monotonically from the base of the root toward the meristem at the root tip. Thus each material point along the root axis is related to a unique time, t, at which it was produced by cell division in the meristem. Instead of its spatial position in an undefinable initial, undeformed state, a material point can be identified by its unique time of production, t. The initial undeformed state can then be defined for each material point individually as the state it was in when it left the meristem. The spatial position of a material point is defined only after its production. Under these assumptions, eqn (1) is modified for a plant root as
Equations (7) and (8) are the mathematical statements of the fact that neighboring material points which were produced next to one another must occupy neighboring spatial locations throughout the growth process. The initial undeformed state can thus be defined for each individual material point as the state it was in at the moment when it was produced by the meristem. We further assume that the relations defined by eqns (5) and (6) are smooth and differentiable over their range of definition. The following quantities are defined under this assumption
x=x[X(t), t], tEt,
(5)
lim [X(x, t)−X(x−dx, t)]=0,
dx 4 0
L'(t, t)= lim
dt 4 0
$
%
x[X(t), t]−x[X(t−dt), t] , dt tEt,
dtq0
(9)
and N(t)=L'(t, t).
(10)
Since a material point, X(t), is at the initial undeformed state at t=t, we have x[X(t), t]−x[X(t−dt), t] =X(t)−X(t−dt)
(11)
at the limit as dt approaches zero. Combining eqns (9) and (11) we define dX(t) =N(t). dt
(12)
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Elongation of the elementary root segment produced at time t is defined as the ratio between its length at time t and its initial length dx[X(t), t] L'(t, t) L(t, t)= = , tEt. dX(t) N(t)
v[X(t),t]= (13)
The quantity N(t) is viewed as the rate of production (mm hr−1 ) at the meristem at time t. After a group of cells is produced at time t, it begins elongation. The elongation function L(t, t) is equal to the ratio between the current length of the cells and their initial length. In general terms, N(t) and L(t, t) are functions of temperature, carbohydrate availability, physical properties of the cell walls, soil water potential, and soil resistance to root penetration. According to eqns (10) and (13), L(t, t)=1.
(14)
Because of the fact that cells grow only to limited length, we have Linf= lim L(t, t)Qa. t4a
(15)
Equations (14) and (15) constitute the most general requirements for the elongation function L. The Henky strain measure used in describing deformation of plastic materials can be used to describe the finite deformation of a plant tissue. The local strain is written for a one-dimensional plant root as: E(t, t)=ln[L(t, t)],
(16)
where E is the local longitudinal strain in the material coordinate system. The rate of deformation, expressed as the time rate of strain, is given by E (t, t)=
1 1L(t, t) 1E(t, t) = . 1t L(t, t) 1t
The velocity at which a material point moves in space is the time derivative of its spatial position
(17)
dx[X(t), t)] = dt
x[X(t), t]=
g
0
1L(g, t) dg (19) 1t
L(t)=
g
t
N(g)L(g, t) dg
(20)
0
and v(t)=
g
t
N(g)
0
1L(g, t) dg+N(t). 1t
(21)
Equations (18)–(21) describe the growth of a plant root as a function of time. From these equations, quantities including the local rate of elongation, the age of root elements at various positions along the root axis, and the process of cell elongation as a function of time can all be derived. This method of identifying the material points makes it possible to consider the growth of each individual cell along the root axis as a function of its own experience of, for example, water stress and carbohydrate availability, as well as age in a dynamic model of root elongation. The validity of the kinematic equations (18)–(21) may be examined by studying the simple case of steady-state root elongation. It is reasonable to assume that the rate of root production in the meristem remains contant under steady-state conditions, i.e. N(t)=N. Equation (21) can then be written as: v(t)=N
g
t
1L(g, t) dg+N. 1t
(22)
Under steady-state conditions, elongation of cells produced at different times would also follow identical trajectories since all cells would be under the same constant environmental conditions. Therefore it may be assumed that L(t, t) of an arbitrary material point to be a function of its age only, so that L(t, t)=L(t−t),
tet.
(23)
Substituting eqns (23) and (14) into eqn (22) yields v(t)=N · L(t),
te0.
(24)
As time progresses, L(t) approaches Linf so that
t
0
N(g)
The total root length and the total rate of root elongation at any moment are described by
0
Alternative terms for the strain rate defined in eqn (17) are the local elementary rate of elongation (Erickson & Sax, 1956b; Gandar, 1983a), or local relative rate of elongation (Sharp et al., 1988). The growth pattern of a root is specified by the meristematic production function, N(t), and the cell elongation function, L(t, t). Both material and spatial coordinate systems can be defined so that the origin is located at the stationary base of the root. Under these definitions, the spatial position of any material point is calculated as
g
t
N(g)L(g, t) dg.
(18)
v=N · Linf for large t,
(25)
where v is the constant, steady-state root elongation rate. These arguments show that under constant environmental conditions, a constant rate of root elongation is achieved when the elongation of the cells produced at the beginning of the root growth process has stopped. The rate of root elongation before this time is transient, even under constant environmental conditions, increasing from v=N at t=0 to the steady-state rate of v=N · Linf . Steady state root growth is characterized by a constant cell size distribution as a function of distance from the root tip (Erickson & Sax, 1956a, b). Cell size and local rate of elongation depend only on the distance from the root tip. Consider an arbitrary point along the root axis which is at a fixed distance from the root tip. As the root grows, older cells at that point are continuously displaced away from the root tip and replaced by arriving younger cells. However, the cell size, age, and rate of elongation remain the same at that point during steady state root growth. The root growth described by eqns (22)–(25) satisfies these requirements. To illustrate this, consider the distance between cells with age a=t−t and the root tip. Under steady state conditions, this is given by d(a)=
g
t
N · L(t−t) dt.
(26)
t−a
where d is distance from root tip. Equation (26) can be written in terms of cell age as: d(a)=
g
a
N · L(a') da',
(27)
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- The relative cell length as a function of distance from the root tip for a corn root during steady-state growth is plotted in Fig. 1(a) using data reported by Erickson & Sax (1956a). The y-axis is the relative cell length calculated as the average cell length along the root axis divided by the average cell length in the meristem. This quantity corresponds to the total local material elongation, L, defined in eqn (13). In calculating L, it is assumed that because this is steady-state growth, the initial length of all cells is the same and equals the cell length in the meristem. The meristem is assumed to be located at 1 mm from the tip of the root, corresponding to the point of minimum cell length and maximum rate of cell division (Erickson & Sax, 1956a, b). The top 1 mm of the root was regarded as the root cap and was not included in the calculations. Cells are at their initial length at the meristem, so that L is equal to 1 at 1 mm from the root tip where the meristem is assumed to be located. The length of the cells increases with increasing distance from the meristem as a result of cell elongation. At a distance of about 8 mm from the root tip, the cell length reaches a constant value as the cells reach maturity. Linf is the ratio between the length of mature cells and their initial length. Figure 1 suggests that 13.0 is a reasonable estimate of Linf . The total rate of root elongation under steady-state conditions reported by Erickson & Sax (1956a) was 1.75 mm hr−1 . Substituting these two values into eqn (25) results in the steady-state root production rate by the meristem as N=0.135 mm hr−1 . Figure 1(a) shows the function L(d) (relative cell length) where d is the distance from root tip. The discussions of theories in the previous section defined
0
which is independent of time and hence satisfies steady-state requirements.
Application: Steady-state Growth of a Corn Root Erickson & Sax (1956a, b) reported detailed measurements of the rate of root elongation, average cell length and local rate of elongation along the axis of a corn root during steady-state growth. Sharp et al. (1988) measured the steady state elongation process of the primary root of maize growing in vermiculite at various water potentials. Pahlavanian & Silk (1988) reported measurements of temperature effects on the elongation of maize primary root. In the following discussion, we apply the theory developed above to the analysis of these experimental observations to illustrate the validity of the theory.
F. 1. Relative cell length of a primary maize root as a function of distance from root tip (a) and cell age (b). The relative cell length is calculated as the ratio between the length of an elongating cell and its initial length (after Erickson & Sax, 1956b).
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L under steady-state conditions as a function of cell age, a [eqn (23)]. Under steady-state conditions, a unique relation d(a) exists [eqn (27)] so that L may be expressed as functions of either a or d. Inversion of eqn (27) produces a(d)=
g
d
0
1 dd. N · L(d)
(28)
This relation is used to convert the distance, d, in Fig. 1 to cell age, a, which is shown in Fig. 1(b), demonstrating that the rate of elongation of a cell accelerates with time over a period of approximately 11 hr. As the cell approaches the maximum size, the rate of elongation quickly decreases to zero. The axial growth of a plant root is usually expressed as the rate of displacement of a material point away from the root tip as a function of distance from the root tip (Erickson & Sax, 1956a, b; List, 1969; Sharp et al., 1988; Bret-Harte & Silk, 1994). This is the rate at which the distance between the root tip and a material point located at d from the root tip increases with time. As d increases, this rate of displacement approaches the rate of total axial root elongation. The rate of displacement between a material point and the root tip under steady-state conditions is calculated by differentiating eqn (23) to yield dd =N · L, dt
(29)
where the relation da/dt=1 is used. The results of calculations using eqn (29) are plotted in Fig. 2 along with the results obtained by measuring displacement of marks on the root surface (Erickson & Sax, 1956b). The calculation agrees well with the results of the measurements except for a small discrepancy near the root tip. This discrepancy is the result of the fact that cell division occurs over a finite segment of the root
F. 2. Measured (W) (Erickson & Sax, 1956a) and predicted (——) rates of displacement from the root tip as a function of distance from the root apex.
while the theory assumes it to occur at an infinitesimal point. These calculations show that the essential information regarding the kinematics of axial root elongation can be evaluated from distribution of cell size along the root axis. It illustrates the ability of the continuum theory in establishing the connection between information from microscopic cell growth studies and those from macroscopic studies of root elongation. Similar use of the continuum theory has also been reported by Silk et al. (1989). Root production in the meristem, N, and the strain rate, E , defined by eqns (10) and (17) correspond to the local growth rates in the meristem and in the elongation region. According to the well-accepted Lockhart theory (Lockhart, 1965; Cleland, 1987) of cell growth, it is reasonable to assume that under steady-state conditions, a constant water stress reduces N and E by constant factors, i.e. NC=b(C) · N0
(30)
E C (t−t)=a(C) · E 0 (t−t),
(31)
and where b(C)E1 and a(C)E1 are functions of soil water potential, N0 and E 0 represent the rate of meristem production and the rate of strain at the reference level of soil water potential. Substituting eqn (31) into eqn (16), and evoking the steady-state condition, results in LC (t−t)=[L0 (t−t)]a .
(32)
Sharp et al. (1988) observed that cessation of longitudinal growth occurred in tissue of approximately the same age regardless of spatial location or water status. This is reflected in eqn (31). The strain rate, E , of a material point under steady-state conditions is a function of its age. When the longitudinal growth of a group of cells stops at a certain age, represented by E =0, eqn (31) indicates that it would stop at the same age for all soil water potentials. The rate of displacement away from the root tip as as function of distance from the root tip under soil water stress is calculated by substituting eqns (30) and (32) into eqn (29). Results of these calculations are shown in Fig. 3. The curve marked ‘‘No stress’’ is the same as the curve shown in Fig. 2, used here as a reference. The curve ‘‘Low stress’’ in Fig. 3 was calculated using a=b=0.9. The curve ‘‘High stress’’ was calculated using a=b=0.8. These values represent a 10% reduction in rates of cell
formation and elongation for ‘‘Low stress’’ and a 20% reduction for ‘‘High stress’’ conditions. These values were chosen so that the reductions in total root elongation rates approximately correspond to those reported by Sharp et al. (1988). These calculations show that a mild 10% reduction in N and E results in a 30% reduction in the rate of steady-state root elongation, and a 20% reduction in N and E may result in a 52% reduction in root elongation rate. Accompanying the reduced root elongation rate is a reduction in the length of the elongation region (Fig. 3). The large reduction in root elongation rate results from the combined effects of lower cell elongation rate and a shorter elongation region. The curves in Fig. 3 agree well with the experimental observations of Sharp et al. (1988) in which the growth curves for the stressed conditions are indistinguishable from that under the no stress condition near the tip of the root. The growth curves for the stressed conditions separate from that of the no stress condition when the stressed cells approach maturity and subsequently stop growing (Fig. 3). By observing the response of the rate of longitudinal displacement along the root axis to water stress (similar to the curves shown in Fig. 3), Sharp et al. (1988) concluded that the inhibition of root elongation at low water potentials was not explained by a constant decrease in growth along the length of the elongation zone. The longitudinal elongation was insensitive to water stress in the early ontogenetic phase of growth but was increasingly inhibited as cells were displaced away from the root tip. Our analysis suggests that experimental observations similar to those shown in Fig. 3 (Sharp et al., 1988; Pritchard et al., 1991) can be explained by assuming that both cell division and elongation are inhibited throughout the elongation region of the root axis by a constant factor. The fact that the rates of longitudinal displacement
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under different water stress conditions are not distinguishable near the root tip may be the result of the manner in which root elongation and cell elongation are related, rather than the evidence that the cells at different ontogenetic phases are affected differently by water stress (Sharp et al., 1988; Pritchard et al., 1991). In addition, our assumption of constant reduction of growth rates throughout the growth region provides a much simpler explanation of the observed root growth patterns under soil water stress (Sharp et al., 1988). The assumption that both the rate of formation and the rate of cell elongation are reduced by a constant factor throughout the growth region is supported by observations of Spollen & Sharp (1991), who reported nearly constant turgor potential in the growth region of corn roots. A decreasing water potential of the growth medium from −0.02 to −1.6 MPa leads to a nearly uniform reduction of root turgor potential from an average of 0.65 MPa to 0.32 MPa. Thus, according to the Lockhart (1965) theory which states that growth is driven by turgor potential in cells, one would expect a constant reduction of growth rates throughout the growth region. - Pahlavanian & Silk (1988) observed that for the primary root of maize, the length of the elongation region was independent of temperature. In the following discussions, we will demonstrate that this is equivalent to stating that a decrease in temperature causes decreases in the rate of biological processes in the root, particularly the cell wall loosening process (Taiz, 1984; Cleland, 1987), so that cells expand at lower rates. However, at the same time, the maturing process may also be slowed by lower temperatures so that cells mature at a later time. In other words, this can simply be stated as the temperature–time equivalence (Sellen, 1980). Assuming that cells in a plant root reach their final length after af hours, af being a function of temperature, the total length of the elongation region under steady-state conditions is calculated by applying eqn (27) as: Le (T )=N(T )
g
af
L(a', T ) da',
(33)
0
F. 3. Predicted effects of water stress on the rates of displacement from root tip. Predictions were made based on the assumption that water stress reduces the steady-state rates of cell formation and elongation by a constant factor.
where Le is the length of the elongation region, a' is the age of the cells, and T is the temperature. In eqn (33), N and L are explicitly expressed as functions of temperature. We are interested in finding the manner in which N, L, and af vary with temperature so that Le is independent of temperature.
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116
We first define an arbitrary reference temperature T0 . The ratio between the length of the elongation region at temperature T and T0 is
Le (T ) N(T ) = Le (T0 ) N(T0 )
g g
af (T)
L(a', T ) da'
0
.
af (T0 )
(34)
L(a', T0 ) da'
0
Letting N(T ) =a(T ), N(T0 )
(35)
the requirement that the length of the elongation region be independent of temperature, i.e. Le (T )= Le (T0 ), dictates that
g g
af (T)
L(a', T ) da'
0 af (T0 )
= L(a', T0 ) da'
1 . a(T )
(36)
0
It can be shown that for an arbitrary function, L(a, T ), eqn (36) is true under the conditions L(a, T )=L[a(T )a, T0 ]
(37)
and af (T )=
af (T0 ) . a(T )
the root tip at three temperatures. The curve marked T0 , which is used as a reference, is the same as the ‘‘No stress’’ curve in Fig. 4, calculated from the data reported by Erickson & Sax (1956a, b). The curves marked T1 and T2 , representing lower temperatures, were calculated using a(T1 )=0.8 and a(T2 )=0.6, respectively. The effects of temperature on root growth shown in Fig. 4 differ from the effects of water stress (Fig. 3) in two significant ways. First, the separation of growth curves at different temperatures occur at the root tip. Second, the rate of total longitudinal root elongation, shown as the rate of displacement of the mature region away from the root tip in Fig. 4, is reduced exactly by the factor of a(T ), thus differing from the effect of water stress where the reduction in the rate of total longitudinal root elongation was significantly higher than the reduction in the rate of cell elongation. In addition, as expected, the length of the elongation region remains constant. Pahlavanian & Silk (1988) measured the response of longitudinal growth of maize primary root to temperature over the range from 16 to 29°C. Their experimental results show exactly the same trend as those shown in Fig. 4, supporting the validity of our theoretical analysis. The principle of temperature–time equivalence states that lower temperatures slow cell elongation. However, elongation of the cells would continue for a longer time so that a constant final cell length is maintained.
(38)
Equations (35), (37) and (38) are the mathematical statements of the temperature–time equivalence concept with respect to plant root elongation. Figure 4 illustrates the rate of displacement of a material point away from the root tip as a function of distance from
F. 4. Predicted effects of temperature on the rates of displacement from the root tip. Curves representing different temperatures were calculated on the basis of the principle of temperature–time equivalence.
Conclusions Kinematic equations describing axial plant root elongation which may serve as the basis of dynamic modeling of the plant root elongation process was derived by considering axial growth of plant roots as consisting of cell formation in the meristem and subsequent elongation. A desirable feature of the theory is the close correspondence between the material description of the root elongation process and the underlying cellular structure, making it possible to compare results of macroscopic observations of root elongation with microscopic observations of cell size distribution along the root axis. The behavior of the kinematic equations under steady-state conditions were examined in detail. The theory was used to examine the effects of soil water stress and temperature on axial root growth process. Experimentally observed axial root growth patterns under soil water stress (Sharp et al., 1988) are explained by a reduction of cell formation and cell elongation process throughout the growth region, consistent with the Lockhart (1965) theory of cell growth and reported effects of soil water
stress on turgor potential distribution in the growth region of the root. Experimentally observed effects of soil temperature on root axial growth are explained by the principle of temperature–time equivalence. This principle may be justified by considering temperature as having a direct effect on biological processes controlling both the loosening of cell walls and the onset of the maturing and termination of the growth, so that cells grow at lower rates but also mature at a longer time at low temperatures. REFERENCES B-H, M. S. & S, W. K. (1994). Nonvascular, symplasmic diffusion of sucrose cannot satisfy the carbon demands of growth in the primary root tip of Zea mays L. Pl. Physiol. 105, 19–33. C, R. E. (1987). The mechanisms of wall loosening and wall extension. In Physiology of Cell Expansion During Plant Growth (Cosgrove, D. J. & Knievel, D. P., eds) pp. 18–27. Am. Soc. Plant Physiol., Pennsylvania State University. E, R. O. & S, K. B. (1956a). Elemental growth rates of the primary root of Zea mays. Proc. Am. phil. Soc. 100, 487–497. E, R. O. & S, K. B. (1956b). Elemental growth rates of the primary root of Zea mays. Proc. Am. Phil. Soc. 100, 499–513. G, P. W. (1983a). Growth in root apices. I. Kinematic description of growth. Bot. Gaz. 144, 1–10.
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G, P. W. (1983b). Growth in root apices. II. Deformation and rate of deformation. Bot. Gaz. 144, 11–19. G, P. B. (1976). Growth and cell pattern formation on an axis. Critique of concepts, terminology and modes of study. Bot. Gaz. 137, 187–202. L, A. J. (1969). Transient growth responses of the primary roots of Zea mays. Planta 87, 1–19. L, J. A. (1965). An analysis of irreversible plant cell elongation. J. theor. Biol. 8, 264–275. MC, E. L. & B, L. (1984). The principles of continuum mechanics applied to transport processes and deformation in plant tissue. J. theor. Biol. 111, 687–705. P, A. M. & S, W. K. (1988). Effect of temperature on spatial and temporal aspects of growth in the primary Maize root. Pl. Physiol. 87, 529–532. P, J. (1991). Turgor, growth and rheological gradients of wheat roots following osmotic stress. J. expl Bot. 42, 1043–1049. S, D. B. (1980). The mechanical properties of plant cell walls. In: The mechanical properties of biological materials. Symp. Soc. expl Biol. 34, 315–329. Cambridge: Cambridge University Press. S, R. E., S, W. K. & H, T. C. (1988). Growth of the Maize primary root at low water potentials. I. Spatial distribution of expansive growth. Pl. Physiol. 87, 50–57. S, W. K. & E, R. O. (1979). Kinematics of plant growth. J. theor. Biol. 76, 481–501. S, W. K., L, E. M. & E, K. J. (1989). Growth patterns inferred from anatomical records. Pl. Physiol. 000, 708–713. S, W. G. & S, R. E. (1991). Spatial distribution of turgor and root growth at low water potentials. Pl. Physiol. 96, 438–443. T, L. (1984). Plant cell expansion: regulation of cell wall mechanical properties. A. Rev. Pl. Physiol. 35, 585–657.
Kinematics of Axial Plant Root Growth Y. F† L. B‡ †Department of Renewable Resources, University of Alberta, Edmonton, Alberta, Canada T6G 2E3 and ‡Department of Crop and Soil Science, Oregon State University, Corvallis, OR 97331, U.S.A. (Received on 19 July 1994, Accepted on 19 October 1994)
Axial growth of plant roots can be modeled as the deformation of a one-dimensional continuum. In this study we present a theory of root elongation that incorporates different roles of cell division in the meristem and axial growth in the elongation region in overall root extension. The meristem is idealized as being located at the root tip, from which material points are produced continuously by cell division. Elongation takes place after material points leave the meristem. Each material point along the root axis is associated with a unique time at which it was produced. The total rate of root growth is the sum of the rate of elongation of all cells present plus the contribution from the meristem. Under steady-state conditions, the rate of elongation of a particular segment of the root is a function of its age only. The theory enables us to compare spatial patterns of steady-state root elongation with distribution of cell sizes along the root axis. The theory is also used to characterize the effects of soil water stress and temperature on root growth. Soil water stress reduces both the rate of meristem production and the subsequent cell elongation, resulting in a shorter elongation region. The effect of temperature is characterized by a temperature–time equivalence effect, leading to an elongation region with its length independent of temperature.
between material and spatial description of growth and the choice of appropriate coordinate systems. This theory has been applied in the analysis of root elongation (Gandar, 1983a, b; McCoy & Boersma, 1984; Sharp et al., 1988; Pahlavanian & Silk, 1988; Silk et al., 1989; Bret-Harte & Silk, 1994). The continuum theory, when applied to plant growth, is concerned essentially with the macroscopic properties of the plant tissue. In describing these properties, we may disregard the fact that a plant tissue must be regarded as discrete on the cellular scale. In attempting to explain these properties, however, it may be necessary to take into account the cellular structure of the plant tissue. Therefore, it is desirable that the material points of continuum theory correspond to particular groups of cells throughout the process of growth, so that comparisons between microscopic scale observations—such as rate of cell elongation and distribution of cell sizes along the root axis—and macroscopic scale observations—such as the rate of deformation—can be made (Silk et al., 1989).
Introduction Axial elongation of plant roots is the result of division and elongation of individual cells. Cell division occurs primarily in the meristem region near the root tip. Cell elongation occurs in an elongation region immediately behind the meristem (Green, 1976). The meristem of the primary root of corn (Zea mays) occupies a region of 1–2 mm long at the root tip. The elongation region extends over a length of about 10 mm from the root meristem (Erickson & Sax, 1956a, b; Sharp et al., 1988; Pahlavanian & Silk, 1988). Silk & Erickson (1979) examined the concept of plant tissue as a mathematical continuum and concluded that for processes involving gross morphological changes, such as cell expansion and division, the continuum assumption is appropriate, although the continuum assumption may not apply for those processes involving individual cell differentiation. They adopted the kinematic theory of conventional continuum mechanics to describe the growth of a plant tissue, and discussed the importance of the distinction 0022–5193/95/090109+09 $08.00/0
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. .
The material points in a continuum are identified by their spatial position in the initial, undeformed, state. The deformation of the continuum is described as the motion of these material points through space. The assumption of non-penetrability of matter states that at any given moment each material point occupies a unique point in space and no two material points may occupy the same spatial position at the same time (Gandar, 1983a). The continuity assumption requires that the neighboring material points remain as neighbors during deformation. Silk & Erickson (1979) and Gandar (1983a, b) presented detailed discussions of these aspects of the continuum mechanics theory and their applications in plant growth studies. These studies established the foundation for a theory which has provided a framework for the quantitative description of growth of plant tissues. However, some fundamental aspects of plant growth remain to be addressed. A central concept of the classical continuum theory is the concept of initial, undeformed state. In a growing root, each cell may be at a different stage of development and deformation. At any given moment, there are cells at the whole spectrum of states ranging from those that have just finished the cell division cycle in the meristem to those that have reached maturity and stopped elongating. Therefore, the concept of the initial, undeformed, state may be applied to individual cells, or material points, as the state they were in at the time they entered the elongation region. It cannot be defined appropriately for a plant root as a whole. The second assumption central to the continuum theory is the equivalence of material and spatial points (Gandar, 1983a). The applicability of this assumption in plant root growth may also be questioned. The spatial position of a particular material point, or a group of cells, could only be defined unambiguously after it has left the meristem. If we regard the material points as having the same spatial position as the meristem before they leave the meristem, i.e. at the root tip, the assumption of material and spatial point equivalence fails since the spatial positions of the material points will not be unique. One may argue that the material points in the continuum theory may not correspond to the cellular structure of the plant root (McCoy & Boersma, 1984). They could be represented by the identifiable points on the surface of the root, such as those identified in streak photography (Erickson & Sax, 1956a; Sharp et al., 1988). This, however, would make it difficult to examine the continuum theory against the results of microscopic observations of cell division and elongation because a material point may represent different groups of cells
at different times. A second consequence of this lack of correspondence between the material points and the cellular structure would be the possibility of infinite elongation of a given material segment. To illustrate this point, examine the root segment between a mark on the tip of the root and a mark at a finite distance from the root tip, such as the base of the root. The elongation of this segment is regarded as a continuous deformation process. Since a root is capable of continued growth, a material segment so defined must be capable of infinite deformation, contradicting the fact that plant cells only grow to a finite size. These observations suggest that the assumption of equivalence between material and spatial points must be modified for plant roots in order to account for the effects of cell division in the meristem and the fact that cells grow to finite sizes. This study was initiated to find a set of simple kinematic equations for plant root growth by modifying the kinematic theory of conventional continuum mechanics. The approach used was to describe the axial growth of a plant root as the result of meristem cell division and subsequent cell elongation. The fact that the axial root growth represents a finite deformation of the plant root as a mathematical continuum is reflected in the resulting equations. In addition, the material points defined in this approach correspond closely to the underlying cellular structures of plant roots, allowing us to compare the results of the continuum theory to those of plant cell growth studies.
Theory In the discussions that follow, we adopt the convention that bold face letters represent vector quantities. Corresponding scalar quantities are denoted by italic letters. Capital letters will be used to represent quantities in material coordinates and corresponding quantities in spatial coordinates will be represented by lower case letters. In the continuum theory, the motion of a typical material point, denoted by X, is expressed as x=x(X, t),
te0,
(1)
where the x on the left-hand side is the spatial position of the material point X at time t; and the x on the right-hand side refers to a function with independent variables X and t. It is assumed that a unique inverse of (1) exists, which is expressed as X=X(x, t),
te0.
(2)
Equation (1) describes the spatial position of a typical material point as a function of time, and eqn (2) describes material points occupying a specific spatial location at different instances in time. Equations (1) and (2) describe the motion of those material points through space during deformation. The assumption of the equivalence of material and spatial points (Gandar, 1983a) is satisfied by the unique inverse relation between eqns (1) and (2). The continuity assumption requires that the neighboring material particles remain being neighbors during deformation. The equivalent mathematical statements of this assumption are lim [x(X+dX, t)−x(X, t)]=0, te0,
(3)
lim [X(x+dx, t)−X(x, t)]=0, te0.
(4)
dX 4 0
and dx 4 0
111
where t identifies the unique time the material point X was produced by the meristem. The condition of non-penetrability of matter is stated for a plant root as: at any moment, tet, an arbitrary material point X(t) occupies a unique spatial position, and no two material points X(t1 ) and X(t2 ), t1$t2 , may occupy the same spatial position at times temax(t1 , t2 ). These statements are equivalent to stating that a unique inverse of (5) exists so that X(t)=X(x, t), tEt,
(6)
The continuity condition, as stated in eqns (3) and (4) for a general continuum, are similarly modified. For axial plant root growth, the continuity condition requires that lim [x[X(t), t]−x[X(t−dt), t]]=0,
dt 4 0
tEt,
dtq0,
(7)
tEt,
dxq0.
(8)
Silk & Erickson (1979) and Gandar (1983a, b) have presented detailed discussions of these aspects of the continuum theory and their applications in plant growth studies.
and
A conceptual process of root growth may be stated as follows. The growth region of a plant root consists of a meristem located at the root tip and an elongation region next to the meristem. Material points, or cells, are produced in the meristem by cell division. Cell elongation occurs after a cell leaves the meristem and continues until it reaches maturity. Both the spatial position and the relative distance between any two material points vary as growth progresses. It is necessary to describe completely the growth pattern of the root in order to identify material points by their spatial position or their position relative to other material points, such as the root tip (Gandar, 1983a). We recognize that the time at which a material point left the meristem increases monotonically from the base of the root toward the meristem at the root tip. Thus each material point along the root axis is related to a unique time, t, at which it was produced by cell division in the meristem. Instead of its spatial position in an undefinable initial, undeformed state, a material point can be identified by its unique time of production, t. The initial undeformed state can then be defined for each material point individually as the state it was in when it left the meristem. The spatial position of a material point is defined only after its production. Under these assumptions, eqn (1) is modified for a plant root as
Equations (7) and (8) are the mathematical statements of the fact that neighboring material points which were produced next to one another must occupy neighboring spatial locations throughout the growth process. The initial undeformed state can thus be defined for each individual material point as the state it was in at the moment when it was produced by the meristem. We further assume that the relations defined by eqns (5) and (6) are smooth and differentiable over their range of definition. The following quantities are defined under this assumption
x=x[X(t), t], tEt,
(5)
lim [X(x, t)−X(x−dx, t)]=0,
dx 4 0
L'(t, t)= lim
dt 4 0
$
%
x[X(t), t]−x[X(t−dt), t] , dt tEt,
dtq0
(9)
and N(t)=L'(t, t).
(10)
Since a material point, X(t), is at the initial undeformed state at t=t, we have x[X(t), t]−x[X(t−dt), t] =X(t)−X(t−dt)
(11)
at the limit as dt approaches zero. Combining eqns (9) and (11) we define dX(t) =N(t). dt
(12)
. .
112
Elongation of the elementary root segment produced at time t is defined as the ratio between its length at time t and its initial length dx[X(t), t] L'(t, t) L(t, t)= = , tEt. dX(t) N(t)
v[X(t),t]= (13)
The quantity N(t) is viewed as the rate of production (mm hr−1 ) at the meristem at time t. After a group of cells is produced at time t, it begins elongation. The elongation function L(t, t) is equal to the ratio between the current length of the cells and their initial length. In general terms, N(t) and L(t, t) are functions of temperature, carbohydrate availability, physical properties of the cell walls, soil water potential, and soil resistance to root penetration. According to eqns (10) and (13), L(t, t)=1.
(14)
Because of the fact that cells grow only to limited length, we have Linf= lim L(t, t)Qa. t4a
(15)
Equations (14) and (15) constitute the most general requirements for the elongation function L. The Henky strain measure used in describing deformation of plastic materials can be used to describe the finite deformation of a plant tissue. The local strain is written for a one-dimensional plant root as: E(t, t)=ln[L(t, t)],
(16)
where E is the local longitudinal strain in the material coordinate system. The rate of deformation, expressed as the time rate of strain, is given by E (t, t)=
1 1L(t, t) 1E(t, t) = . 1t L(t, t) 1t
The velocity at which a material point moves in space is the time derivative of its spatial position
(17)
dx[X(t), t)] = dt
x[X(t), t]=
g
0
1L(g, t) dg (19) 1t
L(t)=
g
t
N(g)L(g, t) dg
(20)
0
and v(t)=
g
t
N(g)
0
1L(g, t) dg+N(t). 1t
(21)
Equations (18)–(21) describe the growth of a plant root as a function of time. From these equations, quantities including the local rate of elongation, the age of root elements at various positions along the root axis, and the process of cell elongation as a function of time can all be derived. This method of identifying the material points makes it possible to consider the growth of each individual cell along the root axis as a function of its own experience of, for example, water stress and carbohydrate availability, as well as age in a dynamic model of root elongation. The validity of the kinematic equations (18)–(21) may be examined by studying the simple case of steady-state root elongation. It is reasonable to assume that the rate of root production in the meristem remains contant under steady-state conditions, i.e. N(t)=N. Equation (21) can then be written as: v(t)=N
g
t
1L(g, t) dg+N. 1t
(22)
Under steady-state conditions, elongation of cells produced at different times would also follow identical trajectories since all cells would be under the same constant environmental conditions. Therefore it may be assumed that L(t, t) of an arbitrary material point to be a function of its age only, so that L(t, t)=L(t−t),
tet.
(23)
Substituting eqns (23) and (14) into eqn (22) yields v(t)=N · L(t),
te0.
(24)
As time progresses, L(t) approaches Linf so that
t
0
N(g)
The total root length and the total rate of root elongation at any moment are described by
0
Alternative terms for the strain rate defined in eqn (17) are the local elementary rate of elongation (Erickson & Sax, 1956b; Gandar, 1983a), or local relative rate of elongation (Sharp et al., 1988). The growth pattern of a root is specified by the meristematic production function, N(t), and the cell elongation function, L(t, t). Both material and spatial coordinate systems can be defined so that the origin is located at the stationary base of the root. Under these definitions, the spatial position of any material point is calculated as
g
t
N(g)L(g, t) dg.
(18)
v=N · Linf for large t,
(25)
where v is the constant, steady-state root elongation rate. These arguments show that under constant environmental conditions, a constant rate of root elongation is achieved when the elongation of the cells produced at the beginning of the root growth process has stopped. The rate of root elongation before this time is transient, even under constant environmental conditions, increasing from v=N at t=0 to the steady-state rate of v=N · Linf . Steady state root growth is characterized by a constant cell size distribution as a function of distance from the root tip (Erickson & Sax, 1956a, b). Cell size and local rate of elongation depend only on the distance from the root tip. Consider an arbitrary point along the root axis which is at a fixed distance from the root tip. As the root grows, older cells at that point are continuously displaced away from the root tip and replaced by arriving younger cells. However, the cell size, age, and rate of elongation remain the same at that point during steady state root growth. The root growth described by eqns (22)–(25) satisfies these requirements. To illustrate this, consider the distance between cells with age a=t−t and the root tip. Under steady state conditions, this is given by d(a)=
g
t
N · L(t−t) dt.
(26)
t−a
where d is distance from root tip. Equation (26) can be written in terms of cell age as: d(a)=
g
a
N · L(a') da',
(27)
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- The relative cell length as a function of distance from the root tip for a corn root during steady-state growth is plotted in Fig. 1(a) using data reported by Erickson & Sax (1956a). The y-axis is the relative cell length calculated as the average cell length along the root axis divided by the average cell length in the meristem. This quantity corresponds to the total local material elongation, L, defined in eqn (13). In calculating L, it is assumed that because this is steady-state growth, the initial length of all cells is the same and equals the cell length in the meristem. The meristem is assumed to be located at 1 mm from the tip of the root, corresponding to the point of minimum cell length and maximum rate of cell division (Erickson & Sax, 1956a, b). The top 1 mm of the root was regarded as the root cap and was not included in the calculations. Cells are at their initial length at the meristem, so that L is equal to 1 at 1 mm from the root tip where the meristem is assumed to be located. The length of the cells increases with increasing distance from the meristem as a result of cell elongation. At a distance of about 8 mm from the root tip, the cell length reaches a constant value as the cells reach maturity. Linf is the ratio between the length of mature cells and their initial length. Figure 1 suggests that 13.0 is a reasonable estimate of Linf . The total rate of root elongation under steady-state conditions reported by Erickson & Sax (1956a) was 1.75 mm hr−1 . Substituting these two values into eqn (25) results in the steady-state root production rate by the meristem as N=0.135 mm hr−1 . Figure 1(a) shows the function L(d) (relative cell length) where d is the distance from root tip. The discussions of theories in the previous section defined
0
which is independent of time and hence satisfies steady-state requirements.
Application: Steady-state Growth of a Corn Root Erickson & Sax (1956a, b) reported detailed measurements of the rate of root elongation, average cell length and local rate of elongation along the axis of a corn root during steady-state growth. Sharp et al. (1988) measured the steady state elongation process of the primary root of maize growing in vermiculite at various water potentials. Pahlavanian & Silk (1988) reported measurements of temperature effects on the elongation of maize primary root. In the following discussion, we apply the theory developed above to the analysis of these experimental observations to illustrate the validity of the theory.
F. 1. Relative cell length of a primary maize root as a function of distance from root tip (a) and cell age (b). The relative cell length is calculated as the ratio between the length of an elongating cell and its initial length (after Erickson & Sax, 1956b).
. .
114
L under steady-state conditions as a function of cell age, a [eqn (23)]. Under steady-state conditions, a unique relation d(a) exists [eqn (27)] so that L may be expressed as functions of either a or d. Inversion of eqn (27) produces a(d)=
g
d
0
1 dd. N · L(d)
(28)
This relation is used to convert the distance, d, in Fig. 1 to cell age, a, which is shown in Fig. 1(b), demonstrating that the rate of elongation of a cell accelerates with time over a period of approximately 11 hr. As the cell approaches the maximum size, the rate of elongation quickly decreases to zero. The axial growth of a plant root is usually expressed as the rate of displacement of a material point away from the root tip as a function of distance from the root tip (Erickson & Sax, 1956a, b; List, 1969; Sharp et al., 1988; Bret-Harte & Silk, 1994). This is the rate at which the distance between the root tip and a material point located at d from the root tip increases with time. As d increases, this rate of displacement approaches the rate of total axial root elongation. The rate of displacement between a material point and the root tip under steady-state conditions is calculated by differentiating eqn (23) to yield dd =N · L, dt
(29)
where the relation da/dt=1 is used. The results of calculations using eqn (29) are plotted in Fig. 2 along with the results obtained by measuring displacement of marks on the root surface (Erickson & Sax, 1956b). The calculation agrees well with the results of the measurements except for a small discrepancy near the root tip. This discrepancy is the result of the fact that cell division occurs over a finite segment of the root
F. 2. Measured (W) (Erickson & Sax, 1956a) and predicted (——) rates of displacement from the root tip as a function of distance from the root apex.
while the theory assumes it to occur at an infinitesimal point. These calculations show that the essential information regarding the kinematics of axial root elongation can be evaluated from distribution of cell size along the root axis. It illustrates the ability of the continuum theory in establishing the connection between information from microscopic cell growth studies and those from macroscopic studies of root elongation. Similar use of the continuum theory has also been reported by Silk et al. (1989). Root production in the meristem, N, and the strain rate, E , defined by eqns (10) and (17) correspond to the local growth rates in the meristem and in the elongation region. According to the well-accepted Lockhart theory (Lockhart, 1965; Cleland, 1987) of cell growth, it is reasonable to assume that under steady-state conditions, a constant water stress reduces N and E by constant factors, i.e. NC=b(C) · N0
(30)
E C (t−t)=a(C) · E 0 (t−t),
(31)
and where b(C)E1 and a(C)E1 are functions of soil water potential, N0 and E 0 represent the rate of meristem production and the rate of strain at the reference level of soil water potential. Substituting eqn (31) into eqn (16), and evoking the steady-state condition, results in LC (t−t)=[L0 (t−t)]a .
(32)
Sharp et al. (1988) observed that cessation of longitudinal growth occurred in tissue of approximately the same age regardless of spatial location or water status. This is reflected in eqn (31). The strain rate, E , of a material point under steady-state conditions is a function of its age. When the longitudinal growth of a group of cells stops at a certain age, represented by E =0, eqn (31) indicates that it would stop at the same age for all soil water potentials. The rate of displacement away from the root tip as as function of distance from the root tip under soil water stress is calculated by substituting eqns (30) and (32) into eqn (29). Results of these calculations are shown in Fig. 3. The curve marked ‘‘No stress’’ is the same as the curve shown in Fig. 2, used here as a reference. The curve ‘‘Low stress’’ in Fig. 3 was calculated using a=b=0.9. The curve ‘‘High stress’’ was calculated using a=b=0.8. These values represent a 10% reduction in rates of cell
formation and elongation for ‘‘Low stress’’ and a 20% reduction for ‘‘High stress’’ conditions. These values were chosen so that the reductions in total root elongation rates approximately correspond to those reported by Sharp et al. (1988). These calculations show that a mild 10% reduction in N and E results in a 30% reduction in the rate of steady-state root elongation, and a 20% reduction in N and E may result in a 52% reduction in root elongation rate. Accompanying the reduced root elongation rate is a reduction in the length of the elongation region (Fig. 3). The large reduction in root elongation rate results from the combined effects of lower cell elongation rate and a shorter elongation region. The curves in Fig. 3 agree well with the experimental observations of Sharp et al. (1988) in which the growth curves for the stressed conditions are indistinguishable from that under the no stress condition near the tip of the root. The growth curves for the stressed conditions separate from that of the no stress condition when the stressed cells approach maturity and subsequently stop growing (Fig. 3). By observing the response of the rate of longitudinal displacement along the root axis to water stress (similar to the curves shown in Fig. 3), Sharp et al. (1988) concluded that the inhibition of root elongation at low water potentials was not explained by a constant decrease in growth along the length of the elongation zone. The longitudinal elongation was insensitive to water stress in the early ontogenetic phase of growth but was increasingly inhibited as cells were displaced away from the root tip. Our analysis suggests that experimental observations similar to those shown in Fig. 3 (Sharp et al., 1988; Pritchard et al., 1991) can be explained by assuming that both cell division and elongation are inhibited throughout the elongation region of the root axis by a constant factor. The fact that the rates of longitudinal displacement
115
under different water stress conditions are not distinguishable near the root tip may be the result of the manner in which root elongation and cell elongation are related, rather than the evidence that the cells at different ontogenetic phases are affected differently by water stress (Sharp et al., 1988; Pritchard et al., 1991). In addition, our assumption of constant reduction of growth rates throughout the growth region provides a much simpler explanation of the observed root growth patterns under soil water stress (Sharp et al., 1988). The assumption that both the rate of formation and the rate of cell elongation are reduced by a constant factor throughout the growth region is supported by observations of Spollen & Sharp (1991), who reported nearly constant turgor potential in the growth region of corn roots. A decreasing water potential of the growth medium from −0.02 to −1.6 MPa leads to a nearly uniform reduction of root turgor potential from an average of 0.65 MPa to 0.32 MPa. Thus, according to the Lockhart (1965) theory which states that growth is driven by turgor potential in cells, one would expect a constant reduction of growth rates throughout the growth region. - Pahlavanian & Silk (1988) observed that for the primary root of maize, the length of the elongation region was independent of temperature. In the following discussions, we will demonstrate that this is equivalent to stating that a decrease in temperature causes decreases in the rate of biological processes in the root, particularly the cell wall loosening process (Taiz, 1984; Cleland, 1987), so that cells expand at lower rates. However, at the same time, the maturing process may also be slowed by lower temperatures so that cells mature at a later time. In other words, this can simply be stated as the temperature–time equivalence (Sellen, 1980). Assuming that cells in a plant root reach their final length after af hours, af being a function of temperature, the total length of the elongation region under steady-state conditions is calculated by applying eqn (27) as: Le (T )=N(T )
g
af
L(a', T ) da',
(33)
0
F. 3. Predicted effects of water stress on the rates of displacement from root tip. Predictions were made based on the assumption that water stress reduces the steady-state rates of cell formation and elongation by a constant factor.
where Le is the length of the elongation region, a' is the age of the cells, and T is the temperature. In eqn (33), N and L are explicitly expressed as functions of temperature. We are interested in finding the manner in which N, L, and af vary with temperature so that Le is independent of temperature.
. .
116
We first define an arbitrary reference temperature T0 . The ratio between the length of the elongation region at temperature T and T0 is
Le (T ) N(T ) = Le (T0 ) N(T0 )
g g
af (T)
L(a', T ) da'
0
.
af (T0 )
(34)
L(a', T0 ) da'
0
Letting N(T ) =a(T ), N(T0 )
(35)
the requirement that the length of the elongation region be independent of temperature, i.e. Le (T )= Le (T0 ), dictates that
g g
af (T)
L(a', T ) da'
0 af (T0 )
= L(a', T0 ) da'
1 . a(T )
(36)
0
It can be shown that for an arbitrary function, L(a, T ), eqn (36) is true under the conditions L(a, T )=L[a(T )a, T0 ]
(37)
and af (T )=
af (T0 ) . a(T )
the root tip at three temperatures. The curve marked T0 , which is used as a reference, is the same as the ‘‘No stress’’ curve in Fig. 4, calculated from the data reported by Erickson & Sax (1956a, b). The curves marked T1 and T2 , representing lower temperatures, were calculated using a(T1 )=0.8 and a(T2 )=0.6, respectively. The effects of temperature on root growth shown in Fig. 4 differ from the effects of water stress (Fig. 3) in two significant ways. First, the separation of growth curves at different temperatures occur at the root tip. Second, the rate of total longitudinal root elongation, shown as the rate of displacement of the mature region away from the root tip in Fig. 4, is reduced exactly by the factor of a(T ), thus differing from the effect of water stress where the reduction in the rate of total longitudinal root elongation was significantly higher than the reduction in the rate of cell elongation. In addition, as expected, the length of the elongation region remains constant. Pahlavanian & Silk (1988) measured the response of longitudinal growth of maize primary root to temperature over the range from 16 to 29°C. Their experimental results show exactly the same trend as those shown in Fig. 4, supporting the validity of our theoretical analysis. The principle of temperature–time equivalence states that lower temperatures slow cell elongation. However, elongation of the cells would continue for a longer time so that a constant final cell length is maintained.
(38)
Equations (35), (37) and (38) are the mathematical statements of the temperature–time equivalence concept with respect to plant root elongation. Figure 4 illustrates the rate of displacement of a material point away from the root tip as a function of distance from
F. 4. Predicted effects of temperature on the rates of displacement from the root tip. Curves representing different temperatures were calculated on the basis of the principle of temperature–time equivalence.
Conclusions Kinematic equations describing axial plant root elongation which may serve as the basis of dynamic modeling of the plant root elongation process was derived by considering axial growth of plant roots as consisting of cell formation in the meristem and subsequent elongation. A desirable feature of the theory is the close correspondence between the material description of the root elongation process and the underlying cellular structure, making it possible to compare results of macroscopic observations of root elongation with microscopic observations of cell size distribution along the root axis. The behavior of the kinematic equations under steady-state conditions were examined in detail. The theory was used to examine the effects of soil water stress and temperature on axial root growth process. Experimentally observed axial root growth patterns under soil water stress (Sharp et al., 1988) are explained by a reduction of cell formation and cell elongation process throughout the growth region, consistent with the Lockhart (1965) theory of cell growth and reported effects of soil water
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