Physicochemical and environmental plant physiology 4th ed p nobel (elsevier, 2009)

Chapter 3 considers solute movement into and out of plant cells, leading to an explanation of electrical potential differences across membranes and establishing the formal criteria for d

Plant Physiology

FOURTH EDITION

Environmental Plant Physiology

FOURTH EDITION

Park S Nobel

Department of Ecology and Evolutionary Biology

University of California, Los Angeles

Los Angeles, California

AMSTERDAM • BOSTON • HEIDELBERG • LONDON

NEW YORK • OXFORD • PARIS • SAN DIEGO

SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO

Academic Press is an imprint of Elsevier

525 B Street, Suite 1900, San Diego, CA 92101-4495, USA

32 Jamestown Road, London NW1 7BY, UK

Fourth Edition 2009

Copyright © 2009, Elsevier Inc All rights reserved

No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher

Permissions may be sought directly from Elsevier’s Science & Technology Rights

Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: permissions@elsevier.com Alternatively you can submit your request online

by visiting the Elsevier web site at http://www.elsevier.com/locate/permissions, and

selecting: Obtaining permission to use Elsevier material

Notice

No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein Because of rapid advances in the medical sciences, in particular, independent verifi cation

of diagnoses and drug dosages should be made

ISBN: 978-0-12-374143-1

For information on all Academic Press publications

visit our website at www.elsevierdirect.com

Printed and bound in Canada

09 10 11 12 10 9 8 7 6 5 4 3 2 1

Preface xiii

2.1E Tensile Strength, Viscosity 54

3.3F Energy for Active Transport 142

3.6A Infl uence of Refl ection Coeffi cients on IncipientPlasmolysis 165

4.4C Absorption Bands, Absorption Coeffi cients,

4.5 Problems 223

5 Photochemistry of Photosynthesis 229

5.4C Photosynthetic Action Spectra and Enhancement Effects 256

6.4 Mitochondrial Bioenergetics 303

7 Temperature and Energy Budgets 319

8.1D Cuticle 376

9.2B Water Vapor 451

Appendix I Numerical Values of Constants and Coeffi cients 545

Index 571

pro-We will specifi cally consider water relations, solute transport, tosynthesis, transpiration, respiration, and environmental interactions A physiologist endeavors to understand such topics in physical and chemical terms; accurate models can then be constructed and responses to the inter-nal and the external environment can be predicted Elementary chemis-try, physics, and mathematics are used to develop concepts that are key to under standing biology—the intent is to provide a rigorous development, not a compendium of facts References provide further details, although in some cases the enunciated principles carry the reader to the forefront of current research Calculations are used to indicate the physiological conse-quences of the various equations, and problems at the end of chapters pro-vide fur ther such exercises Solutions to all of the problems are provided, and the appendixes have a large list of values for constants and conversion factors at various temperatures.

pho-Chapters 1 through 3 describe water relations and ion transport for plant cells In Chapter 1, after discussing the concept of diffusion, we consider the physical barriers to diffusion imposed by cellular and organelle membranes Another physical barrier associated with plant cells is the cell wall, which limits cell size In the treatment of the movement of water through cells in response to specifi c forces presented in Chapter 2, we employ the thermody-namic argument of chemical potential gradients Chapter 3 considers solute movement into and out of plant cells, leading to an explanation of electrical potential differences across membranes and establishing the formal criteria for distinguishing diffusion from active transport Based on concepts from irreversible thermodynamics, an important parameter called the refl ection coeffi cient is derived, which permits a precise evaluation of the infl uence of osmotic pressures on fl ow

The next three chapters deal primarily with the interconversion of ious forms of energy In Chapter 4 we consider the properties of light and

var-its absorption After light is absorbed, var-its radiant energy usually is rapidly converted to heat However, the arrangement of photosynthetic pigments and their special molecular structures allow some radiant energy from the sun to be converted by plants into chemical energy In Chapter 5 we dis-cuss the particular features of chlorophyll and the accessory pigments for photosynthesis that allow this energy conversion Light energy absorbed by chloroplasts leads to the formation of ATP and NADPH These compounds represent currencies for carrying chemical and electrical (redox potential) energy, respectively How much energy they actually carry is discussed in Chapter 6.

In the last three chapters we consider the various forms in which energy and matter enter and leave a plant as it interacts with its environment The physical quantities involved in an energy budget analysis are presented in Chapter 7 so that the relative importance of the various factors affecting the temperature of leaves or other plant parts can be quantitatively evaluated The resistances (or their reciprocals, conductances) affecting the movement

of both water vapor during transpiration and carbon dioxide during synthesis are discussed in detail for leaves in Chapter 8, paying particular attention to the individual parts of the pathway and to fl ux density equa-tions The movement of water from the soil through the plant to the atmo-sphere is discussed in Chapter 9 Because these and other topics depend

photo-on material introduced elsewhere in the book, the text is extensively referenced

cross-This text is the fourth edition of Physicochemical and Environmental Plant Physiology (Academic Press, 3rd ed., 2005; 2nd ed., 1999; 1st ed., 1991),

which evolved from Biophysical Plant Physiology and Ecology (Freeman, 1983), Intro duction to Biophysical Plant Physiology (Freeman, 1974), and Plant Cell Physiology: A Physicochemical Approach (Freeman, 1970) The

text has been updated based on the ever-increasing quality of plant research

as well as comments of colleagues and students The goal is to integrate the physical sciences, engineering, and mathematics to help understand biology,

especially for plants Physicochemical and Environmental Plant Physiology,

is suitable for existing situations and habitats as well as new applications

Park S Nobel October 20, 2008

Symbols and

Abbreviations

Where appropriate, typical units are indicated in parentheses

chl superscript for chloroplast

hydrogen electrode (mV)

G giga (as a prefix), 109

Lsoil soil hydraulic conductivity coefficient (m2Pa1s1)

(flux density per unit force)

in grams; contains Avogadro’s number of molecules

Pj permeability coefficient of species j (m s1)

orbital

UB minimum kinetic energy to cross barrier (J mol1)

fraction of an electronic charge

thickness of air boundary layer (mm)

an absorption band or for the maximum photon (or energy)emission in an emission spectrum

such an orbital

p* an excited or antibonding electron orbital in a molecule or

an electron in such an orbital

(dimensionless)

Plant Physiology

FOURTH EDITION

1 Cells and Diffusion

1.1 Cell Structure 31.1A Generalized Plant Cell 31.1B Leaf Anatomy 51.1C Vascular Tissue 71.1D Root Anatomy 9

1.2 Diffusion 111.2A Fick's First Law 121.2B Continuity Equation and Fick's Second Law 141.2C Time–Distance Relation for Diffusion 161.2D Diffusion in Air 19

1.3 Membrane Structure 211.3A Membrane Models 211.3B Organelle Membranes 23

1.4 Membrane Permeability 251.4A Concentration Difference Across a Membrane 261.4B Permeability Coefficient 281.4C Diffusion and Cellular Concentration 29

1.5 Cell Walls 311.5A Chemistry and Morphology 331.5B Diffusion Across Cell Walls 341.5C Stress–Strain Relations of Cell Walls 371.5D Elastic Modulus, Viscoelasticity 39

1.1A Generalized Plant Cell

illus-trates the larger subcellular structures The living material of a cell, known asthe protoplast, is surrounded by the cell wall The cell wall is composed ofcellulose and other polysaccharides, which helps provide rigidity to

3

individual cells as well as to the whole plant The cell wall containsnumerous relatively large interstices, so it is not the main permeabilitybarrier to the entry of water or small solutes into plant cells The mainbarrier, known as the plasma membrane (or plasmalemma), is foundinside the cell wall and surrounds the cytoplasm The permeability ofthis membrane varies with the particular solute, so the plasma membranecan regulate what enters and leaves a plant cell The cytoplasm containsorganelles such as chloroplasts and mitochondria, which are membrane-surrounded compartments in which energy can be converted from oneform to another Chloroplasts, whose production and maintenance is aprimary function of plants, are the sites for photosynthesis, and mito-chondria are the sites for respiration Microbodies, such as peroxisomesand ribosomes, are also found in the cytoplasm along with macromole-cules and other structures that influence the thermodynamic properties

of water Thus, the term cytoplasm includes the organelles (but generallynot the nucleus), whereas the term cytosol refers to the cytoplasmicsolution delimited by the plasma membrane and the tonoplast (to bediscussed next) but exterior to the organelles

In mature cells of higher (evolutionarily advanced) plants and manylower plants, there is a large central aqueous compartment, the centralvacuole, which is surrounded by a membrane called the tonoplast Thecentral vacuole is usually quite large and can occupy up to about 90% ofthe volume of a mature cell Because of the large central vacuole, the

area across which diffusion can occur The aqueous solution in the centralvacuole contains mainly inorganic ions or organic acids as solutes, althoughconsiderable amounts of sugars and amino acids may be present in somespecies Water uptake by this central vacuole occurs during cell growth andhelps lead to the support of a plant

Chloroplasts

VacuolePeroxisomes

Figure 1-1 Schematic representation of a mature mesophyll cell from the leaf of a flowering plant,

suggest-ing some of the complexity resultsuggest-ing from the presence of many membrane-surrounded cellular compartments.

sub-One immediate impression of plant cells is the great prevalence ofmembranes In addition to surrounding the cytoplasm, membranes alsoseparate various compartments in the cytoplasm Diffusion of substancesacross these membranes is much more difficult than is diffusion withinthe compartments Thus, organelle and vacuolar membranes can controlthe contents and consequently the reactions occurring in the particularcompartments that they surround Diffusion can also impose limitations

on the overall size of a cell because the time for diffusion can increasewith the square of the distance, as we will quantitatively consider in thenext section

Although many plant and algal cells share most of the features indicated

inFigure 1-1, they are remarkably diverse in size The cells of the green alga

some species of the intertidal green alga Valonia have multinucleated cells

as large as 20 mm in diameter The genera Chara and Nitella include and brackish-water green algae having large internodal cells (Fig 3-13) thatmay be 100 mm long and 1 mm in diameter Such large algal cells haveproved extremely useful for studying ion fluxes, as we consider inChapter 3 (e.g., Sections 3.2E; 3.3E,F)

fresh-1.1B Leaf Anatomy

A cross section of a typical angiosperm (seed plant) leaf can illustratevarious cell types and anatomical features that are important for photosyn-thesis and transpiration Leaves are generally 4 to 10 cells thick, which

on both the upper and the lower sides of a leaf and is usually one cell layerthick Except for the guard cells, epidermal cells usually are colorless be-cause their cytoplasm contains few, if any, chloroplasts (depending on thespecies) Epidermal cells have a relatively thick waterproof cuticle on the

diverse group of complex polymers composed principally of esters of 16- and18-carbon monocarboxylic acids that have two or three hydroxyl groups(esterification refers to the chemical joining of an acid and an alcoholresulting in the removal of a water molecule) Cutin is relatively inert andalso resists enzymatic degradation by microorganisms, so it is often wellpreserved in fossil material We will consider its role in minimizing waterloss from a leaf

Between the two epidermal layers is the mesophyll (literally, middle ofthe leaf) tissue, which is usually differentiated into chloroplast-containing

“palisade” and “spongy” cells The palisade cells are often elongated pendicular to the upper epidermis and are found immediately beneath it

meso-phyll cells and the lower epidermis, are loosely packed, and intercellular airspaces are conspicuous In fact, most of the surface area of both spongy andpalisade mesophyll cells is exposed to air in the intercellular spaces, facili-tating diffusion of gases into or out of the cells A spongy mesophyll cell is

40 chloroplasts (AsFig 1-2illustrates, the cells are by no means cally regular, so dimensions here indicate only approximate size.) A neigh-

spongy mesophyll cell can have a volume of

is called a stoma or stomate (plural: stomata and stomates, respectively)

of water vapor by transpiration also occurs mainly through the stomatalpores, as we will discuss in Chapter 8 (Section 8.1B) Stomata thus serve

Intercellularair space

100μmChloroplasts

Lowerepidermis

Spongymesophyllcells

Palisademesophyllcells

CuticleUpperepidermis

Figure 1-2 Schematic transverse section through a leaf, indicating the arrangement of various cell types.

Often about 30 to 40 mesophyll cells occur per stoma.

as a control, helping to strike a balance between freely admitting the CO2needed for photosynthesis and at the same time preventing excessive loss of

stomata

1.1C Vascular Tissue

The xylem and the phloem make up the vascular systems found contiguously

constitutes a layer of the bark and the xylem constitutes almost all of thewood The xylem provides structural support for land plants Water conduc-

which lies just inside the vascular cambium (region of meristematic activityfrom which xylem and phloem cells differentiate) Outside the functioningphloem are other phloem cells that can be shed as pieces of bark slough off.Phloem external to the xylem, as in a tree, is the general pattern for the stems

of plants As we follow the vascular tissue from the stem along a petiole andinto a leaf, we observe that the xylem and the phloem often form a vein,which sometimes conspicuously protrudes from the lower surface of a leaf.Reflecting the orientation in the stem or the trunk, the phloem is foundabaxial to the xylem in the vascular tissue of a leaf (i.e., the phloem is located

on the side of the lower epidermis) The vascular system branches andrebranches as it crosses a dicotyledonous leaf, becoming smaller (in crosssection) at each step In contrast to the reticulate venation in dicotyledons,monocotyledons characteristically have parallel-veined leaves Individualmesophyll cells in the leaf are never further than a few cells from the vasculartissue

The movement of water and nutrients from the soil to the upper portions

of a plant occurs primarily in the xylem The xylem sap usually contains

nitro-gen that are metabolically produced in the root The xylem is a tissue ofvarious cell types that we will consider in more detail in the final chapter(Section 9.4B,D), when water movement in plants is discussed quantitative-

ly The conducting cells in the xylem are the narrow, elongated tracheids andthe vessel members (also called vessel elements), which tend to be shorter andwider than the tracheids Vessel members are joined end-to-end in long

1 The rings in trees are not always annual In many desert species a ring forms when large xylem cells are produced after a suitable rainy period followed by smaller cells, and this can occur more than once or sometimes not at all in a particular year Moreover, trees from the wet tropics can have no annual rings.

2 Molarity (moles of solute per liter of solution, symbolized by M) is a useful unit for concentration, but it is not recommended for highly accurate measurements by the international unit conven- tion, Le Syst
eme International d’Unites or Syst
eme International (SI) Nevertheless, we will use molarity in addition to the SI unit of mol m3 We also note that SI as currently practiced allows the American spelling “liter” and “meter” as well as the British spelling “litre” and “metre.”

linear files; their adjoining end walls or perforation plates have from onelarge hole to many small holes The conducting cells lose their protoplasts,and the remaining cell walls thus form a low-resistance channel for thepassage of solutions Xylem sap moves from the root, up the stem, throughthe petiole, and then to the leaves in these hollow dead xylem “cells,” withmotion occurring in the direction of decreasing hydrostatic pressure Somesolutes leave the xylem along the stem on the way to a leaf, and others diffuse

or are actively transported across the plasma membranes of various leaf cellsadjacent to the conducting cells of the xylem

The movement of most organic compounds throughout the plant takesplace in the other vascular tissue, the phloem A portion of the photosyn-thetic products made in the mesophyll cells of the leaf diffuses or is activelytransported across cellular membranes until it reaches the conducting cells

of the leaf phloem By means of the phloem, the photosynthetic products—which then are often mainly in the form of sucrose—are distributed through-out the plant The carbohydrates produced by photosynthesis and certainother substances generally move in the phloem toward regions of lower

Sieve-tubemember

Sieveplate

Companioncell

Vesselmember

Fibercell

Sievetube

Figure 1-3 Idealized longitudinal section through part of a vascular bundle in a stem, illustrating various

anatomical aspects of the xylem and the phloem New cells forming in the xylem initially contain cytoplasm, which is lost as the cells mature and become conducting Fiber cells, which occur in the xylem, are usually quite tapered and provide structural support The nucleated companion cells are metabolically involved with the sieve-tube members of the phloem.

concentration, although diffusion is not the mechanism for the movement, asindicated in Chapter 9 (Section 9.4F,G) The phloem is a tissue consisting ofseveral types of cells In contrast to the xylem, however, the conducting cells

of the phloem contain cytoplasm They are known as sieve cells and

system throughout the plant Although these phloem cells often contain nonuclei at maturity, they remain metabolically active Cells of the phloem,including companion cells, are further discussed in Chapter 9 (Section 9.4E)

1.1D Root Anatomy

Roots anchor plants in the ground as well as absorb water and nutrientsfrom the soil and then conduct these substances upward to the stem.Approximately half of the products of photosynthesis are allocated toroots for many plants To help understand uptake of substances into aplant, we will examine the cell types and the functional zones that occuralong the length of a root

relatively undifferentiated cells that are scraped off as the root grows intonew regions of the soil Cell walls in the root cap are often mucilaginous,which can reduce friction with soil particles Proximal to the root cap is ameristematic region where the cells rapidly divide Cells in this apical mer-istem tend to be isodiametric and have thin cell walls Next is a region of cellelongation in the direction of the root axis Such elongation mechanicallypushes the root tip through the soil, causing cells of the root cap to slough off

by abrasion with soil particles Sometimes the region of dividing cells is not

Root cap

Pericycle Phloem

Cambium Xylem

Endodermis Cortex Epidermis

(a) Figure 1-4 Schematic diagrams of a young root: (a) longitudinal section, indicating the zones that can occur

near the root tip; and (b) cross-sectional view approximately 10 mm back from the tip, ing the arrangement of the various cell types.

indicat-spatially distinct from the elongation zone Also, cell size and the extent ofthe zones vary with both plant species and physiological status.

where the cells begin to assume more highly specialized functions The cellwalls become thicker, and elongation is greatly diminished The epidermalcells develop fine projections, radially outward from the root, called roothairs These root hairs greatly increase the surface area across which waterand nutrients can enter a plant As we follow a root toward the stem, the rootsurface generally becomes less permeable to water and the root interiorbecomes more involved with conducting water toward the stem Watermovement into the root is discussed in Chapter 9 (Section 9.4A), so thediscussion here is restricted to some of the morphological features

The region of a young root where water absorption most readily occurs

root at the level where root hairs are found Starting from the outside, weobserve first the root epidermis and then a number of layers of cells known

as the cortex There are abundant intercellular air spaces in the cortex,

generally are lacking in vascular tissue) Inside the cortex is a single layer ofcells, the endodermis The radial and transverse walls of the endodermal cellsare impregnated with waxy material, including suberin, forming a band

passage of water and solutes across that part of the cell wall Because thereare no air spaces between endodermal cells, and the radial walls are blocked

by the waterproof Casparian strip, water must pass through the lateral wallsand enter the cytoplasm of endodermal cells to continue across a root Theendodermal cells can represent the only place in the entire pathway forwater movement from the soil, through the plant, to the air where it is

pathway, water can move in cell walls or in the hollow lumens of xylemvessels, a region referred to as the apoplast

Immediately inside the endodermis is the pericycle, which is typicallyone cell thick in angiosperms The cells of the pericycle can divide and form ameristematic region that can produce lateral or branch roots in the regionjust above the root hairs Radially inside the pericycle is the vascular tissue.The phloem generally occurs in two to eight or more strands located aroundthe root axis The xylem usually radiates out between the phloem strands, sowater does not have to cross the phloem to reach the xylem of a young root.The tissue between the xylem and the phloem is the vascular cambium,which through cell division and differentiation produces xylem (to the inside

in stems and older roots) and phloem (to the outside in stems and olderroots)

3 In the roots of many species a subepidermal layer or layers of hypodermis occur Radial walls of hypodermal cells can also be blocked with a waxy material analogous to the Casparian strip in the endodermis, in which case the layers are often termed an exodermis.

Our rather elementary discussion of leaves, vascular tissues, and rootsleads to the following oversimplified but useful picture The roots take upwater from the soil along with nutrients required for growth These areconducted in the xylem to the leaves Leaves of a photosynthesizing plant

phloem back to the root Thus, the xylem and the phloem serve as the

“plumbing” that connects the two types of plant organs that are functionallyinteracting with the environment To understand the details of such physi-ological processes we must turn to fields like calculus, physics, thermody-namics, and photochemistry Our next step is to bring the abstract ideas ofthese fields into the realm of cells and plants, which means that we need tomake calculations using appropriate assumptions and approximations

We begin the text by describing diffusion (Chapter 1) To discuss water(Chapter 2) and solutes (Chapter 3), we will introduce the thermodynamicconcept of chemical potential This leads to a quantitative description offluxes, electrical potentials across membranes, and the energy requirementsfor active transport of solutes Some important energy conversion processestake place in the organelles For instance, light energy is absorbed(Chapter 4) by photosynthetic pigments located in the internal membranes

of chloroplasts (Chapter 5) and then converted into other forms of energyuseful to a plant (Chapter 6) or dissipated as heat (Chapter 7) Leaves(Chapter 8) as well as groups of plants (Chapter 9) also interact with the

problem-solving approach to these topics, we will pay particular attention to sions and ranges for the parameters as well as to the insights that can begained by developing the relevant formulae and then making calculations

dimen-1.2 Diffusion

Diffusion leads to the net movement of a substance from a region of higherconcentration to an adjacent region of lower concentration of that substance(Fig 1-5) It is a spontaneous process; that is, no energy input is required

Net flux

Figure 1-5 The random thermal motion of uncharged molecules of species j produces a net movement from

a region of higher concentration (left-hand side) to a region of lower concentration (right-hand side).

Diffusion takes place in both the liquid and the gas phases associated withplants and is a result of the random thermal motion of the molecules—thesolute(s) and the solvent in the case of a solution or of gases in the case of air.The net movement caused by diffusion is a statistical phenomenon—a great-

er probability exists for molecules to move from the concentrated region to

per unit volume are present in the concentrated region than in the dilute one,

so more are available for diffusing toward the dilute region than are able for movement in the opposite direction If left isolated from externalinfluences, diffusion of a neutral species tends to even out concentrationdifferences originally present in adjoining regions of a liquid or a gas In fact,the randomizing tendency of the generally small, irregular motion of parti-cles by diffusion is a good example of the increase in entropy, or disorder,that accompanies all spontaneous processes In 1905, Albert Einstein de-scribed such diffusion as a case of Brownian motion or movement, whichwas first observed microscopically by Robert Brown in 1827 for colloidalparticles

avail-Diffusion is involved in many plant processes, such as gas exchangeand the movement of nutrients toward root surfaces For instance, diffu-

atmosphere up to the leaf surface and then diffuses through the stomatal

membrane of a leaf mesophyll cell, and then diffuses through the cytosol

diffuses up to the enzymes that are involved in carbohydrate formation

not have to be by diffusion, refinements that we will return to inChapter 8, Section 8.3D) In this chapter we develop the mathematicalformulation necessary for understanding both diffusion across a mem-brane and diffusion in a solution

1.2A Fick’s First Law

solute species j in a solution or gaseous species j in air; the subscript jindicates that we are considering only one species of the many that could

be present We will assume that the concentration of species j in some region

is less than in a neighboring one A net migration of molecules occurs by

speak-ing, this applies to neutral molecules or in the absence of electrical potentialdifferences, an aspect that we will return to in Chapter 3, Section 3.2) Such amolecular flow down a concentration gradient is analogous to the flow of

heat from a warmer to a cooler region The analogy is actually good cially for gases) because both processes depend on the random thermalmotion of molecules In fact, the differential equations and their solutionsthat are used to describe diffusion are those that had previously been devel-oped to describe heat flow.

(espe-To express diffusion quantitatively, we will consider a diffusive flux orflow of species j For simplicity, we will restrict our attention to diffusioninvolving planar fronts of uniform concentration, a situation that has wide-

amount of species j crossing a certain area per unit time, for example, moles

Reasoning by analogy with heat flow, Fick deduced that the “force,” orcausative agent, leading to the net molecular movement involves the con-centration gradient A gradient indicates how a certain parameter changeswith distance; the gradient in concentration of species j in the x-direction is

changes as we move a short distance along the x-axis when other variables,such as time and position along the y-axis, are held constant In general, theflux density of some substance is proportional to an appropriate force, arelation that we will use repeatedly in this text In the present instance,the driving force is the negative of the concentration gradient of species j,

appreciate why a negative sign occurs, recall that the direction of net itive) diffusion is toward regions of lower concentration We can now writethe following relation showing the dependence of the flux density on thedriving force:

the negative concentration gradient into an equality

Equation 1.1 is commonly known as Fick’s first law of diffusion,

for diffusion, it is properly called a coefficient in the general case Incertain applications, however, we can obtain sufficient accuracy by treat-

indicate the change in concentration in the x-direction of Cartesiancoordinates at some moment in time (constant t) and for specified values

of y and z For most of the cases that we will consider, the flux density in

4 Although the SI convention recommends the term flux density, much of the diffusion literature refers to Jjas a flux Moreover, many symbols have been used for flux density (e.g., A, D, E, F, I, J,

M, Q, U, and V), some of which (such as A for assimilation and E for evaporation) conflict with those used for other common variables (A for area and E for electric field or potential) We have chosen J because of its lack of conflict and the long precedent for its use (e.g., Lars Onsager used J for the flux densities of heat and mass in the early 1930s).

the x-direction has the same magnitude at any value of y and z, meaningthat we are dealing with one-dimensional, planar fluxes By convention, anet flow in the direction of increasing x is positive (from left to right in

concentra-tion, we again note that the negative sign is needed in Equation 1.1.Fick’s first law indicates that diffusion is greater when the concentrationgradient is steeper or the diffusion constant is larger It has been amplydemonstrated experimentally and is the starting point for our formaldiscussion of diffusion

1.2B Continuity Equation and Fick’s Second Law

As we indicated earlier, diffusion in a solution is important for themovement of solutes across plant cells and tissues How rapid are suchprocesses? For example, if we release a certain amount of material in onelocation, how long will it take before we can detect that substance atvarious distances? To discuss such phenomena adequately, we must de-termine the dependence of the concentration on both time and distance

We can readily derive such a time–distance relationship if we first sider the conservation of mass, which is necessary if we are to transformEquation 1.1 into an expression that is convenient for describing theactual solute distributions caused by diffusion In particular, we want

on x and t

The amount of solute or gaseous species j per unit time crossing agiven area, here considered to be a planar area perpendicular to the x-axis

J j+ J j dx x

Figure 1-6 Diagram showing the dimensions and the flux densities that form the geometric basis for the

continuity equation The same general figure is used to discuss water flow in Chapter 2 (Section 2.4F) and solute flow in Chapter 3 (Section 3.3A).

(Fig 1-6), can change with position along the x-axis Let us imagine avolume element of thickness dx in the direction of flow and of cross-

multiplied by the distance, dx, gives the overall change in the flux density,

Adx in unit time for this one-dimensional case is the amount flowing

species j divided by the volume Thus, the change in the amount of species

j in the volume element in unit time can also be expressed as the change in

which the change in concentration occurs, Adx Equating these two differentexpressions that describe the rate of change in the amount of species j in thevolume Adx, we obtain the following relation:

Then after division through by Adx, Equation 1.2 leads to the very usefulexpression known as the continuity equation:

The continuity equation is a mathematical way of stating that mattercannot be created or destroyed under ordinary conditions Thus, if the flux

Equation 1.3 indicates that its concentration must be increasing with time, asthe material is then accumulating locally If we substitute Fick’s first law (Eq.1.1) into the continuity equation (Eq 1.3), we obtain Fick’s second law For

2cj

the following expression for the concentration of species j satisfies the

(t = 0) placed in a plane located at the origin of the x-direction (i.e., at x = 0,whereas y and z can have any value, which defines the plane considered

concentra-tion funcconcentra-tion that depends on posiconcentra-tion and time as given by Eq 1.5 and the

ide-alized situation can be achieved by inserting a radioactive tracer in a plane atthe origin of the x-direction Equation 1.5 is only one of the possible solutions

to the second-order partial differential equation representing Fick’s secondlaw The form is relatively simple compared with other solutions, and, moreimportant, the condition of having a finite amount of material released at aparticular location is realistic for certain applications to biological problems

1.2C Time–Distance Relation for Diffusion

partic-ular solution to Fick’s second law (Eq 1.4) and is restricted to the case of

understand-ing diffusion It relates the distance a substance diffuses to the time sary to reach that distance The expression uses the diffusion coefficient of

Equation 1.5 indicates that the concentration in the plane at the origin of

Þ, which becomesinfinitely large as t is turned back to 0, the starting time Practically speaking,

the solute initially placed as close as possible to a plane at the origin For tgreater than 0, the solute diffuses away from the origin The distribution of

5 To show that Equation 1.5 is a possible solution of Fick’s second law, it can be substituted into Equation 1.4 and the differentiations performed (Mj and Dj are constant; Lax n =Lx ¼ anxn1; Le axn

=Lx ¼ anx n1eaxn

; and Luv=Lx ¼ uLv=Lx þ vLu=LxÞ The solution of Equation 1.4 becomes progressively more difficult when more complex conditions or molecular interactions (which cause variations in Dj) are considered Indeed, many books have been written on the solutions to Equation 1.4, where Equation 1.5 actually represents the first term of a power series [note that ( pDjt) 1/2 can be replaced by ffiffiffiffiffiffiffiffiffiffi

pD jt p ].

concentration profiles along the time axis Because the total amount of

i.e., in a volume element parallel to the x-axis and extending from x values of

¥ to +¥), the area under each of the concentration profiles is the same

of the diffusing molecules from the origin increases with time Also,

time increases, as the diffusing solute or gaseous species is then distributedover a greater region of space In estimating how far molecules diffuse in

drops to 1/e or 37% of its value in the plane at the origin Although what arbitrary, this parameter describes the shift of the statistical

(d)(c) t

t

x x

Figure 1-7 Concentration of species j, cj, as a function of position x for molecules diffusing according to

Fick’s second law The molecules were initially placed in a plane at the origin of the x-direction, that is, at x = 0 For a given value of x, cj is the same throughout a plane in the y- and the z- directions (a) Distribution of concentrations along the x-axis occurring at a time ta, (b) distri- bution occurring at a subsequent time tb, (c) portrayal of the concentration profiles at ta and tb, and (d) three-dimensional surface portraying change of concentration with time and position Note that x1/eis the location at which the concentration of species j has dropped to 1/e of its value

at the origin.

distribution of the population of molecules with time From Equation 1.5,

Equation 1.6 is an extremely important relationship that indicates afundamental characteristic of diffusion processes: The distance a population

of molecules of a particular solute or gaseous species diffuses—for the dimensional case in which the molecules are released in a plane at the origin

one-—is proportional to the square root of both the diffusion coefficient of thatspecies and the time for diffusion In other words, the time to diffuse a givendistance increases with the square of that distance An individual molecule

origin; that is, we are dealing with the characteristics of a whole population

of molecules, not the details of an individual molecule Furthermore, thefactor 4 is rather arbitrary because some criterion other than 1/e causes thisnumerical factor to be somewhat different, although the basic form of Equa-tion 1.6 is preserved For example, the numerical factor is (ln 2)(4) or 2.8 ifthe criterion is to drop to half of the value at the origin

Table 1-1lists the magnitudes of diffusion coefficients for various solutes

the greater frictional interaction between water molecules and fibrous teins than with the more compact globular ones, fibrous proteins often havediffusion coefficients that are approximately half of those of globular pro-teins of the same molecular weight

pro-To illustrate the time–distance consequences of Equation 1.6, wequantitatively consider the diffusion of small molecules in an aqueous

6 The symbolC in this text represents degrees on the Celsius temperature scale By the SI system, the Celsius degree as well as the kelvin unit or a kelvin (abbreviated K) is 1/273.16 of the thermodynamic temperature of the triple point of water (0.01000C) and absolute zero is at

273.15 C The term “centigrade” is no longer recommended.

7 Relative molecular mass, an expression that is preferred over the commonly used “molecular weight,” is a dimensionless number indicating the molecular mass of a substance relative to that

of a neutral carbon atom with six protons and six neutrons (12C) taken as 12.00000 For proteins, the molecular mass is often expressed in kilodaltons (kDa), where 1 Da is 1

12 the mass of12C For instance, sucrose has a relative molecular mass, or molecular weight, of 342 and a molecular mass

of 342 Da.