Planta (Berl.) 113, 355--366 (1973) 9 by Springer-Verlag 1973
On the Volume-flow Mechanism of Phloem Transport J o h n H. Young, R a y F. E v e r t , a n d W a l t e r Eschrich School of Pharmacy, Theoretical Chemistry Institute, and Department of Botany, University of Wisconsin, Madison, Wisconsin 53706, USA, and Forstbotanisches Institut der Universit/it, 1)-3400 GSttingen, Federal Republic of Germany Received May 25, 1973
Summary. A steady-state model of solution flow in a tubular semipermeable membrane is developed for an arbitrary distribution of solute sources and sinks along the translocation path. I t is demonstrated that the volume-flow mechanism of phloem transport depends only on the two assumptions: 1. that the plasmalemma of the sieve tube is a differentially permeable membrane, and 2. that sugars are actively secreted into and absorbed from the lumen of the sieve tube. I t is shown that in the absence of a pressure gradient, there is a negligible concentration gradient over most of the translocation path. However, in the presence of a pressure gradient a small concentration gradient develops as a result of the continually changing chemical potential of water along the direction of solution flow. For Poiseuille flow the concentration gradient is approximately proportional to the mean stream velocity. Introduction W e r e c e n t l y carried o u t a s t u d y of solution flow t h r o u g h t u b u l a r s e m i p e r m e a b l e m e m b r a n e s as a model for p h l o e m t r a n s p o r t (Eschrich et al., 1972). T h e results of t h a t s t u d y were a n a l y z e d in t e r m s of a simple m a t h e m a t i c a l m o d e l in which t h e d r i v i n g force of solution flow is t h e difference in t h e chemical p o t e n t i a l of w a t e r across t h e s e m i p e r m e a b l e m e m b r a n e , a n d f a i r l y good q u a n t i t a t i v e results were o b t a i n e d . W e d e m o n s t r a t e d t h a t t h e p h y s i c a l basis of solution flow is t h e same in Miinch's m o d e l s y s t e m (1930) as in ours, a n d t h a t solution flow would occur in b o t h s y s t e m s in t h e absence of a pressure g r a d i e n t . Of course, pressure g r a d i e n t s arising from resistance to viscous flow exist in a n y real system, b u t these pressure gradients, irrespective of t h e i r m a g n i t u d e , are in no w a y a n essential f e a t u r e of solution flow in these m o d e l systems. To use t h e t e r m " p r e s s u r e f l o w " in t h e sense originally used b y Miineh t o describe solution flow in these s y s t e m s would, therefore, be a misnomer. Consequently, we p r o p o s e d t h e t e r m " v o l u m e f l o w " as a more appro-
356
J.H. Young et al.
priate designation for solution flow both in these model systems and in the phloem. I n the model systems, solution flow occurs as the consequence of an initial nonequilibrium system relaxing toward osmotic equilibrium. I n the plant the system can be maintained in a nonequilibrium state b y the active secretion and absorption of solutes into and out of the lumen of the sieve tube. I f the rates of secretion and absorption are only slowly varying fmlctions of time, the system would tend to a steady-state condition. Nonetheless, the same tendency of the system to relax to a state of osmotic equilibrium would produce solution flow. I t is now well known t h a t parenehymatous elements anywhere along the translocation path can act either as sources or sinks of solutes, depending on conditions (Crafts and Crisp, 1971). To obtain a realistic model of phloem transport one must, therefore, eonsidcr the problem of solution flow in a tubular semipermeable membrane under steady state conditions for an arbitrary distribution of sources and sinks. This will allow one to determine how the distribution and strength of the sources and sinks determine the direction and rate of solution flow. Mass flow in the sense used here is essentially the displacement of an incompressible fluid due to the influx or efflux of volume at the boundaries of the system. I n particular, it should be noted that the hydrodynamic equations of motion based on Newton's second law are not required for rationalizing the basic phenomenon of solution flow. To put it another way, neither the inertial nor the viscous properties of the fluid are intrinsic to the basic phenomenon of solution flow. I t is for this reason t h a t the t e r m " v o l u m e flow" is a more appropriate designation of the mechanism than the t e r m "pressure flow". However, some eonfusion has arisen on this point. I n a recent comm e n t a r y on the volume-flow mechanism, Weatherley (1973) incorrectly asserts that the t e r m " v o l u m e flow" implies "flow along sieve tubes when the pressure gradient involved is very small". As should be clear from the preceeding discussion, and as we a t t e m p t e d to make clear previously, apparently unsuccessfully, the t e r m "volume flow" has nothing whatsoever to do with the magnitude of the pressure gradient. Weatherley also states that we interpreted our results " i n terms of the mechanism put forward by Miinch" and t h a t our model "involves no new concepts and does not constitute any mechanism that is not implicit in the Mfineh t h e o r y " . I t would be more correct to say that our model would have been implicit in a precise interpretation of Miinch's experiment had Mfinch's experiment been more precisely interpreted prior to our analysis. The difference between the volume-flow and pressure-flow mechanisms is merely the difference between a precise and an imprecise interpretation of one and the same phenomenon. Concerning the matter
Volume-flow Mechanism of Phloem Transport
357
of viscous flow the point to be emphasized is t h a t this effect cannot be properly handled until the physical basis of solution flow is correctly perceived and the appropriate formalism developed. We will, therefore, begin by treating in the present paper the case of steady state solution flow in a closed semipermeable tube with an arbitrary distribution of solute sources and sinks along its length in the absence of a pressure gradient. Once the qualitative behavior of this system is understood, it is a relatively straightforward m a t t e r to include the effect of a pressure gradient arising from resistance to viscous flow. We will show t h a t for viscous solution flow through a closed, tubular semipermeable membrane the stream velocity far from a source or sink is proportional to the solute concentration gradient. From this effect one can estimate the magnitude of the apparent "diffusion coefficient" reported by Mason and Maskell (1928).
Steady-state Solution Flow Let us consider a closed semipermeable tube of length b and radius r. Neglecting diffusion and considering only convection, the rate of change of solute flux with position along the axis of the tube under steady state conditions is given by the equation d azr2~7 x (civ)=2azr N (x) (1) where x denotes the distance from one end of the tube, ci(x ) the local concentration inside the tube, v (x) the mean stream velocity, and N (x) the flux of solute through the walls of the tube. I n a source region N (x) > 0, whereas in a sink region N (x) < 0. The boundary conditions for a closed tube are v (o)= v (b)= 0. Using the condition v (o)= 0 and integrating Eq. (1) yields
q(x) v(x)=M(x)
(2)
where X
M 0 (influx) for c i > g,i and J ( x ) < 0 (efflux) for c i < gi, as can be seen by using Eq. (6) to rewrite J ( x ) (5) as J ( x ) = Lp R T (vi---di).
(8)
Volume-flow Mechanism of Phloem Transport
359
Solving Eq. (8) for c~ and using Eq. (4) yields
st (x) = ~i(1 + ~v')
(9)
where v'=dv/clx and r
T= 2Lplie q
(10)
is the same as the relaxation time for the movement of the concentration front in ore" model system [Eq. (13) of Eschrich et al., 1972]. Substituting Eq. (9) into Eq. (2) yields a first order differential equation for v(x) v~ i (1@ ~ v ' ) : M (x).
(11)
For ~ v ' ~ 1, the solution of Eq. (11) is simply v(x)~ M(x)/g i. Thus v (x) is everywhere proportional to M(x) in this limit. Comparing this result with Eq. (2) implies that c i (x)~ ci for all x, as can also be seen from Eq. (9). Thus there would be virt~ally no concentration gradient accompanying solution flow under these conditions. The physical interpretation of this result is clear. For T sufficiently small the system relaxes to osmotic equilibrium faster than it can be perturbed from equilibrium by the eontinnal secretion and absorption of solutes at sources and sinks. Consequently, the solution inside the tube is approximately in a state of osmotic equilibrium everywhere along the tube, and local infinitesimal departures from osmotic equilibrium are sufficient to maintain a nonzero stream velocity. For the more general case in which the inequality ~ v ' ~ 1 is not satisfied, one can still determine the qualitative behavior of v(x). Iris useful to use Eq. (2) and rewrite Eq. (11) as v ' = T - 1 ( ~ - 1).
(12)
In the neighborhood of a source ci> gi so that v'> 0. Even beyond the region where the solute is actively secreted, v(x) increases as long as c i > g i. However, as v(x) increases due to the influx of water, ci(x ) decreases for the same reason and approaches gi as an asymptotic limit. As ei(x ) approaches -di, v'(x) approaches zero and v(x) approaches a constant value in a region far from a source or sink. If the sources and sinks were localized at the ends of the tube, there would be no significant concentration gradient over most of the translocation path. I n the neighborhood of a sink c i < g ~ so that v'
On the Volume-flow Mechanism of Phloem Transport J o h n H. Young, R a y F. E v e r t , a n d W a l t e r Eschrich School of Pharmacy, Theoretical Chemistry Institute, and Department of Botany, University of Wisconsin, Madison, Wisconsin 53706, USA, and Forstbotanisches Institut der Universit/it, 1)-3400 GSttingen, Federal Republic of Germany Received May 25, 1973
Summary. A steady-state model of solution flow in a tubular semipermeable membrane is developed for an arbitrary distribution of solute sources and sinks along the translocation path. I t is demonstrated that the volume-flow mechanism of phloem transport depends only on the two assumptions: 1. that the plasmalemma of the sieve tube is a differentially permeable membrane, and 2. that sugars are actively secreted into and absorbed from the lumen of the sieve tube. I t is shown that in the absence of a pressure gradient, there is a negligible concentration gradient over most of the translocation path. However, in the presence of a pressure gradient a small concentration gradient develops as a result of the continually changing chemical potential of water along the direction of solution flow. For Poiseuille flow the concentration gradient is approximately proportional to the mean stream velocity. Introduction W e r e c e n t l y carried o u t a s t u d y of solution flow t h r o u g h t u b u l a r s e m i p e r m e a b l e m e m b r a n e s as a model for p h l o e m t r a n s p o r t (Eschrich et al., 1972). T h e results of t h a t s t u d y were a n a l y z e d in t e r m s of a simple m a t h e m a t i c a l m o d e l in which t h e d r i v i n g force of solution flow is t h e difference in t h e chemical p o t e n t i a l of w a t e r across t h e s e m i p e r m e a b l e m e m b r a n e , a n d f a i r l y good q u a n t i t a t i v e results were o b t a i n e d . W e d e m o n s t r a t e d t h a t t h e p h y s i c a l basis of solution flow is t h e same in Miinch's m o d e l s y s t e m (1930) as in ours, a n d t h a t solution flow would occur in b o t h s y s t e m s in t h e absence of a pressure g r a d i e n t . Of course, pressure g r a d i e n t s arising from resistance to viscous flow exist in a n y real system, b u t these pressure gradients, irrespective of t h e i r m a g n i t u d e , are in no w a y a n essential f e a t u r e of solution flow in these m o d e l systems. To use t h e t e r m " p r e s s u r e f l o w " in t h e sense originally used b y Miineh t o describe solution flow in these s y s t e m s would, therefore, be a misnomer. Consequently, we p r o p o s e d t h e t e r m " v o l u m e f l o w " as a more appro-
356
J.H. Young et al.
priate designation for solution flow both in these model systems and in the phloem. I n the model systems, solution flow occurs as the consequence of an initial nonequilibrium system relaxing toward osmotic equilibrium. I n the plant the system can be maintained in a nonequilibrium state b y the active secretion and absorption of solutes into and out of the lumen of the sieve tube. I f the rates of secretion and absorption are only slowly varying fmlctions of time, the system would tend to a steady-state condition. Nonetheless, the same tendency of the system to relax to a state of osmotic equilibrium would produce solution flow. I t is now well known t h a t parenehymatous elements anywhere along the translocation path can act either as sources or sinks of solutes, depending on conditions (Crafts and Crisp, 1971). To obtain a realistic model of phloem transport one must, therefore, eonsidcr the problem of solution flow in a tubular semipermeable membrane under steady state conditions for an arbitrary distribution of sources and sinks. This will allow one to determine how the distribution and strength of the sources and sinks determine the direction and rate of solution flow. Mass flow in the sense used here is essentially the displacement of an incompressible fluid due to the influx or efflux of volume at the boundaries of the system. I n particular, it should be noted that the hydrodynamic equations of motion based on Newton's second law are not required for rationalizing the basic phenomenon of solution flow. To put it another way, neither the inertial nor the viscous properties of the fluid are intrinsic to the basic phenomenon of solution flow. I t is for this reason t h a t the t e r m " v o l u m e flow" is a more appropriate designation of the mechanism than the t e r m "pressure flow". However, some eonfusion has arisen on this point. I n a recent comm e n t a r y on the volume-flow mechanism, Weatherley (1973) incorrectly asserts that the t e r m " v o l u m e flow" implies "flow along sieve tubes when the pressure gradient involved is very small". As should be clear from the preceeding discussion, and as we a t t e m p t e d to make clear previously, apparently unsuccessfully, the t e r m "volume flow" has nothing whatsoever to do with the magnitude of the pressure gradient. Weatherley also states that we interpreted our results " i n terms of the mechanism put forward by Miinch" and t h a t our model "involves no new concepts and does not constitute any mechanism that is not implicit in the Mfineh t h e o r y " . I t would be more correct to say that our model would have been implicit in a precise interpretation of Miinch's experiment had Mfinch's experiment been more precisely interpreted prior to our analysis. The difference between the volume-flow and pressure-flow mechanisms is merely the difference between a precise and an imprecise interpretation of one and the same phenomenon. Concerning the matter
Volume-flow Mechanism of Phloem Transport
357
of viscous flow the point to be emphasized is t h a t this effect cannot be properly handled until the physical basis of solution flow is correctly perceived and the appropriate formalism developed. We will, therefore, begin by treating in the present paper the case of steady state solution flow in a closed semipermeable tube with an arbitrary distribution of solute sources and sinks along its length in the absence of a pressure gradient. Once the qualitative behavior of this system is understood, it is a relatively straightforward m a t t e r to include the effect of a pressure gradient arising from resistance to viscous flow. We will show t h a t for viscous solution flow through a closed, tubular semipermeable membrane the stream velocity far from a source or sink is proportional to the solute concentration gradient. From this effect one can estimate the magnitude of the apparent "diffusion coefficient" reported by Mason and Maskell (1928).
Steady-state Solution Flow Let us consider a closed semipermeable tube of length b and radius r. Neglecting diffusion and considering only convection, the rate of change of solute flux with position along the axis of the tube under steady state conditions is given by the equation d azr2~7 x (civ)=2azr N (x) (1) where x denotes the distance from one end of the tube, ci(x ) the local concentration inside the tube, v (x) the mean stream velocity, and N (x) the flux of solute through the walls of the tube. I n a source region N (x) > 0, whereas in a sink region N (x) < 0. The boundary conditions for a closed tube are v (o)= v (b)= 0. Using the condition v (o)= 0 and integrating Eq. (1) yields
q(x) v(x)=M(x)
(2)
where X
M 0 (influx) for c i > g,i and J ( x ) < 0 (efflux) for c i < gi, as can be seen by using Eq. (6) to rewrite J ( x ) (5) as J ( x ) = Lp R T (vi---di).
(8)
Volume-flow Mechanism of Phloem Transport
359
Solving Eq. (8) for c~ and using Eq. (4) yields
st (x) = ~i(1 + ~v')
(9)
where v'=dv/clx and r
T= 2Lplie q
(10)
is the same as the relaxation time for the movement of the concentration front in ore" model system [Eq. (13) of Eschrich et al., 1972]. Substituting Eq. (9) into Eq. (2) yields a first order differential equation for v(x) v~ i (1@ ~ v ' ) : M (x).
(11)
For ~ v ' ~ 1, the solution of Eq. (11) is simply v(x)~ M(x)/g i. Thus v (x) is everywhere proportional to M(x) in this limit. Comparing this result with Eq. (2) implies that c i (x)~ ci for all x, as can also be seen from Eq. (9). Thus there would be virt~ally no concentration gradient accompanying solution flow under these conditions. The physical interpretation of this result is clear. For T sufficiently small the system relaxes to osmotic equilibrium faster than it can be perturbed from equilibrium by the eontinnal secretion and absorption of solutes at sources and sinks. Consequently, the solution inside the tube is approximately in a state of osmotic equilibrium everywhere along the tube, and local infinitesimal departures from osmotic equilibrium are sufficient to maintain a nonzero stream velocity. For the more general case in which the inequality ~ v ' ~ 1 is not satisfied, one can still determine the qualitative behavior of v(x). Iris useful to use Eq. (2) and rewrite Eq. (11) as v ' = T - 1 ( ~ - 1).
(12)
In the neighborhood of a source ci> gi so that v'> 0. Even beyond the region where the solute is actively secreted, v(x) increases as long as c i > g i. However, as v(x) increases due to the influx of water, ci(x ) decreases for the same reason and approaches gi as an asymptotic limit. As ei(x ) approaches -di, v'(x) approaches zero and v(x) approaches a constant value in a region far from a source or sink. If the sources and sinks were localized at the ends of the tube, there would be no significant concentration gradient over most of the translocation path. I n the neighborhood of a sink c i < g ~ so that v'
