Physicochemical and Environmental Plant Physiology

By Park S. Nobel

Physicochemical and Environmental Plant Physiology, Fifth Edition, is the updated version of an established and successful text and reference for plant scientists. This work represents the seventh book in a 50-year series by Park Nobel beginning in 1970. The original structure and philosophy of the book continue in this new edition, providing a genuine synthesis of modern physicochemical and physiological thinking, while updating the content. Key concepts in plant physiology are developed with the use of chemistry, physics, and mathematics fundamentals.The book contains plant physiology basics while also including many equations and often their derivation to quantify the processes and explain why certain effects and pathways occur, helping readers to broaden their knowledge base. New topics included in this edition are advances in plant hydraulics, other plant–water relations, and the effects of climate change on plants. This series continues to be the gold standard in environmental plant physiology.

• Describes the chemical and the physical principles behind plant physiological processes
• Provides key equations for each chapter and solutions for the problems on each topic
• Includes features that enhances the utility of the book for self-study such as problems after each chapter and the 45-page section "Solution to Problems" at the end of the book
• Includes appendices with conversation factors, constants/coefficients, abbreviations, and symbols

New to this edition:

• The scientific fields and the nationalities of the more than 115 scientists mentioned in the book, providing a nice personal touch
• While adding over 100 new or updated references, reference of special importance historically are retained, showing how science has advanced over the ages
• The often challenging problems at the end of each chapter provide an important test of the mastery of the topics covered. Moreover, the solutions to the problems are presented in detail at the end of the book. The book can thus be used in courses but also especially useful for students or other persons studying this often difficult material on their own
• Finally and most important, the fifth edition continues the emphasis of a quantitative approach begun fifty years ago by Park Nobel (1970) with the publication of his first book in the series. Over the next fifty years from 1970 to 2020, the author has gained considerable experience on how to present quantitative and often abstract material to students. This edition is most likely the final version in the series, which not only covers some of his unique contributions but also has helped countless students and colleagues appreciate the power and insight gained into biology from calculations!

• Language: English
• Publisher: Academic Press
• Release date: Jan 7, 2020
• ISBN: 9780128191477

Park S. Nobel

Park S. Nobel is the Distinguished Professor of Biology Emeritus in the Department of Ecology and Evolutionary Biology at the University of California, Los Angeles. His early career focused on cell physiology, especially chloroplasts, and his first book was entitled Plant Cell Physiology: A Physicochemical Approach (W.H. Freeman, 1970). He eventually shifted toward plant physiological ecology and has written six books on the subject that have been cited extensively. Besides writing these texts, he has published six books on agaves and cacti. He has also authored nearly 400 scientific research articles and reviews. Dr. Nobel has developed original equations for the air boundary layers surrounding cylinders and spheres and has championed the importance of the mesophyll surface area per unit leaf area, Ames/A. Other research topics have included the importance of shallow root distribution for taking advantage of light desert rainfalls and the influences of an air gap developing around roots during drought on root-soil water movement.

Book preview

Physicochemical and Environmental

Plant Physiology

Fifth Edition

Park S. Nobel

Department of Ecology and Evolutionary Biology, University of California, Los Angeles, Los Angeles, California

Table of Contents

Cover image

Title page

Copyright

Preface

Symbols and Abbreviations

Major Equations

1. Cells and Diffusion

1.1. Cell Structure

1.2. Diffusion

1.3. Membrane Structure

1.4. Membrane Permeability

1.5. Cell Walls

1.6. Summary

1.7. Problems

Major Equations

2. Water

2.1. Physical Properties

2.2. Chemical Potential

2.3. Central Vacuole and Chloroplasts

2.4. Water Potential and Plant Cells

2.5. Summary

2.6. Problems

Major Equations

3. Solutes

3.1. Chemical Potential of Ions

3.2. Fluxes and Diffusion Potentials

3.3. Characteristics of Crossing Membranes

3.4. Mechanisms for Crossing Membranes

3.5. Principles of Irreversible Thermodynamics

3.6. Solute Movement Across Membranes

3.7. Summary

3.8. Problems

Major Equations

4. Light

4.1. Wavelength and Energy

4.2. Absorption of Light by Molecules

4.3. Deexcitation

4.4. Absorption Spectra and Action Spectra

4.5. Summary

4.6. Problems

Major Equations

5. Photochemistry of Photosynthesis

5.1. Chlorophyll—Chemistry and Spectra

5.2. Other Photosynthetic Pigments

5.3. Excitation Transfers Among Photosynthetic Pigments

5.4. Groupings of Photosynthetic Pigments

5.5. Electron Flow

5.6. Summary

5.7. Problems

Major Equations

6. Bioenergetics

6.1. Gibbs Free Energy

6.2. Biological Energy Currencies

6.3. Chloroplast Bioenergetics

6.4. Mitochondrial Bioenergetics

6.5. Energy Flow in the Biosphere

6.6. Summary

6.7. Problems

Major Equations

7. Temperature and Energy Budgets

7.1. Energy Budget—Radiation

7.2. Heat Conduction and Convection

7.3. Latent Heat—Transpiration

7.4. Further Examples of Energy Budgets

7.5. Soil

7.6. Summary

7.7. Problems

Major Equations

8. Leaves and Fluxes

8.1. Resistances and Conductances—Transpiration

8.2. Water Vapor Fluxes Accompanying Transpiration

8.3. CO2 Conductances and Resistances

8.4. CO2 Fluxes Accompanying Photosynthesis

8.5. Water-Use Efficiency

8.6. Summary

8.7. Problems

Major Equations

9. Plants and Fluxes

9.1. Gas Fluxes Above Plant Canopy

9.2. Gas Fluxes Within Plant Communities

9.3. Water Movement in Soil

9.4. Water Movement in the Xylem and the Phloem

9.5. Soil–Plant–Atmosphere Continuum

9.6. Global Climate Change

9.7. Summary

9.8. Problems

Solutions To Problems

Appendix I. Numerical Values of Constants and Coefficients

Appendix II. Conversion Factors and Definitions

Appendix III. Mathematical Relations

Appendix IV. Gibbs Free Energy and Chemical Potential

Index

Copyright

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Preface

Let us begin with some comments on the title. Physiology, which is the study of the function of cells, organs, and organisms, derives from the Latin physiologia, which in turn comes from the Greek physi- or physio-, a prefix meaning natural, and logos meaning reason or thought. Thus physiology suggests natural science and is now a branch of biology dealing with processes and activities that are characteristic of living things. Physicochemical relates to physical and chemical properties, and Environmental refers to topics such as solar irradiation and wind. Plant indicates the main focus of this book, but the approach, equations developed, and appendices apply equally well to animals and other organisms.

We will specifically consider water relations, solute transport, photosynthesis, 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 internal and the external environment can be predicted. Elementary chemistry, physics, and mathematics are used to develop concepts key to understanding 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 indicate the physiological consequences of the various equations, and problems at the end of chapters provide further 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.

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 specific forces presented in Chapter 2, we employ the thermodynamic 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 reflection coefficient is derived, which permits a precise evaluation of the influence of osmotic pressures on flow.

The next three chapters deal primarily with the interconversion of various forms of energy. In Chapter 4, we consider the properties of light and its absorption. After light is absorbed, 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 discuss 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 photosynthesis are discussed for leaves in Chapter 8, paying particular attention to the individual parts of the pathway. The movement of water from the soil through the plant to the atmosphere is discussed in Chapter 9 together with new comments on global climate change. Because these and other topics depend on material introduced elsewhere in the book, the text is extensively cross-referenced.

This text is the fifth edition (©2020) of Physicochemical and Environmental Plant Physiology (Academic Press/Elsevier, 4th ed., 2009; 3rd ed., 2005; 2nd ed., 1999; 1st ed., 1991), which evolved from Biophysical Plant Physiology and Ecology (Freeman, 1983), Introduction to Biophysical Plant Physiology (Freeman, 1974), and Plant Cell Physiology: A Physicochemical Approach (Freeman, 1970). Thus the series has encompassed 50 years!

This new edition includes recent plant research and over 100 new or updated references while retaining historically important articles. The text now presents the nationalities plus scientific fields of the 115 scientific pioneers specifically mentioned. Summaries are added to each chapter in response to requests by students. But the real inspiration for this new edition is to instill appreciation for the future uses of physics, chemistry, engineering, and mathematics to help understand biology, especially for plants. Physicochemical and Environmental Plant Physiology, 5th ed., thus continues a tradition to emphasize fundamentals plus a quantitative approach suitable for existing situations and habitats as well as for new applications.

Park S. Nobel

November 4, 2019

Symbols and Abbreviations

Where appropriate, typical units are indicated in parentheses.

---

a The activity, aj, is often considered to be dimensionless, in which case the activity coefficient, γj, has the units of reciprocal concentration (aj  =  γjcj; Eq. 2.5).

b We note that mol liter−¹, or molarity (M), is a concentration unit of widespread use, although it is not an SI unit.

Major Equations

Fick’s first law  (1.1)

Solution to Fick’s second law  (1.5)

Time—distance relation for diffusion  (1.6)

Permeability coefficient  (1.9)

Elastic property  (1.14)

1

Cells and Diffusion

1.1. Cell Structure

1.1A. Generalized Plant Cell

1.1B. Leaf Anatomy

1.1C. Vascular Tissue

1.1D. Root Anatomy

1.2. Diffusion

1.2A. Fick’s First Law

1.2B. Continuity Equation and Fick’s Second Law

1.2C. Time–Distance Relation for Diffusion

1.2D. Diffusion in Air

1.3. Membrane Structure

1.3A. Membrane Models

1.3B. Organelle Membranes

1.4. Membrane Permeability

1.4A. Concentration Difference Across a Membrane

1.4B. Permeability Coefficient

1.4C. Diffusion and Cellular Concentration

1.5. Cell Walls

1.5A Chemistry and Morphology

1.5B. Diffusion Across Cell Walls

1.5C. StresseStrain Relations of Cell Walls

1.5D. Elastic Modulus, Viscoelasticity

1.6. Summary

1.7. Problems

1.8. References and Further Reading

1.1. Cell Structure

Cells are the basis of life! Before formally considering diffusion and related topics, we will therefore outline the structures of certain plant cells and tissues. This will introduce most of the anatomical terms used throughout the book.

1.1A. Generalized Plant Cell

Figure 1-1 depicts a representative leaf cell from a higher plant and illustrates the larger subcellular structures. The living material of a cell, known as the protoplast, is surrounded by the cell wall. The cell wall is composed of cellulose and other polysaccharides, which helps provide rigidity to individual cells as well as to the whole plant. The cell wall contains numerous relatively large interstices (porous regions), so it is not the main permeability barrier to the entry of water or small solutes into plant cells. The main barrier, known as the plasma membrane (or plasmalemma), is found inside the cell wall and surrounds the cytoplasm. The permeability of this membrane varies with the particular solute, so the plasma membrane can regulate what enters and leaves a plant cell.

The cytoplasm contains organelles such as chloroplasts and mitochondria, which are membrane-surrounded compartments in which energy can be converted from one form to another. Chloroplasts, whose production and maintenance is a primary function of plants, are the sites for photosynthesis, and mitochondria are the sites for respiration. Microbodies, such as peroxisomes and ribosomes, are also found in the cytoplasm along with macromolecules and other structures that influence the thermodynamic properties of water. Thus, the term cytoplasm includes the organelles (but generally not the nucleus), whereas the term cytosol refers to the cytoplasmic solution delimited by the plasma membrane and the tonoplast (to be discussed next) but exterior to the organelles.

In mature cells of higher (evolutionarily more advanced) plants and many lower plants, there is a large central aqueous compartment, the central vacuole, which is surrounded by a membrane called the tonoplast. The central vacuole is usually quite large and can occupy up to about 90% of the volume of a mature cell. Because of the large central vacuole, the cytoplasm occupies a thin layer around the periphery of a plant cell (Fig. 1-1). Therefore, for its volume, the cytoplasm has a relatively large surface area across which diffusion can occur. The aqueous solution in the central vacuole contains mainly inorganic ions or organic acids as solutes, although considerable amounts of sugars and amino acids may be present in some species. Water uptake by this central vacuole occurs during cell growth and helps lead to the support of a plant.

Figure 1-1  Schematic representation of a mature mesophyll cell from the leaf of a flowering plant, suggesting some of the complexity resulting from the presence of many membrane-surrounded subcellular compartments.

One immediate impression of plant cells is the great prevalence of membranes. In addition to surrounding the cytoplasm, membranes also separate various compartments in the cytoplasm. Diffusion of substances across these membranes is much more difficult than is diffusion within the compartments. Thus, organelle and vacuolar membranes can control the contents and consequently the reactions occurring in the particular compartments that they surround. Diffusion can also impose limitations on the overall size of a cell because the time for diffusion can increase with the square of the distance, as we will quantitatively consider in the next section.

Although many plant and algal cells share most of the features indicated in Figure 1-1, they are remarkably diverse in size. The cells of the green alga Chlorella are approximately 4  ×  10−⁶  m (4  μm) in diameter. In contrast, some species of the intertidal green alga Valonia have multinucleated cells as large as 20  mm in diameter. The genera Chara and Nitella include fresh- and brackish-water green algae having large internodal cells (Fig. 3-13) that may be 100  mm long and 1  mm in diameter. Such large algal cells have proved extremely useful for studying ion fluxes, as we consider in Chapter 3 (e.g., Sections 3.2E; 3.3E,F).

1.1B. Leaf Anatomy

A cross section of a typical angiosperm (seed plant) leaf can illustrate various cell types and anatomical features that are important for photosynthesis and transpiration. Leaves are generally 4 to 10 cells thick, which corresponds to a few hundred micrometers (Fig. 1-2). An epidermis occurs on both the upper and the lower sides of a leaf and is usually one cell layer thick. Except for the guard cells, epidermal cells usually are colorless because their cytoplasm contains few, if any, chloroplasts (depending on the species). Epidermal cells have a relatively thick waterproof cuticle on the atmospheric side (Fig. 1-2). The cuticle contains cutin, which consists of a diverse group of complex polymers composed principally of esters of 16- and 18-carbon monocarboxylic acids that have two or three hydroxyl groups (esterification refers to the chemical joining of an acid and an alcohol resulting in the removal of a water molecule). Cutin is relatively inert and also resists enzymatic degradation by microorganisms, so it is often well preserved in fossil material. We will consider its role in minimizing water loss from a leaf (e.g., Chapter 7).

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.

Between the two epidermal layers is the mesophyll (literally, middle of the leaf) tissue, which is usually differentiated into chloroplast-containing palisade and spongy cells. The palisade cells are often elongated perpendicular to the upper epidermis and are found immediately beneath it (Fig. 1-2). The spongy mesophyll cells, located between the palisade mesophyll cells and the lower epidermis, are loosely packed, and intercellular air spaces are conspicuous. In fact, most of the surface area of both spongy and palisade mesophyll cells is exposed to air in the intercellular spaces, facilitating diffusion of gases into or out of the cells (Fig. 8-4). A spongy mesophyll cell is often rather spherical, about 20  μm in radius, and can contain approximately 40 chloroplasts. (As Fig. 1-2 illustrates, the cells are by no means geometrically regular, so dimensions here indicate only approximate size.) A neighboring palisade cell is usually more oblong; it can be 80  μm long, can contain 60 chloroplasts, and might be represented by a cylinder 15  μm in radius and 50  μm long with hemispherical ends. Based on the dimensions given, a (spherical) spongy mesophyll cell can have a volume of

A palisade mesophyll cell can have a volume of

(Formulas for areas and volumes of various geometric shapes are given in Appendix IIIB.) In many leaves, palisade mesophyll cells contain about 70% of the chloroplasts and often outnumber the spongy mesophyll cells nearly two to one.

The pathway of least resistance for gases to cross an epidermis—and thus to enter or to exit from a leaf—is through the adjustable space between a pair of guard cells (Fig. 1-2). This pore, and its two surrounding guard cells, is called a stoma or stomate (plural: stomata and stomates, respectively). When they are open, the stomatal pores allow for the entry of CO2 into the leaf and for the exit of photosynthetically produced O2. The inevitable loss of water vapor by transpiration also occurs mainly through the stomatal pores, as we will discuss in Chapter 8 (Section 8.1B). Stomata thus serve as a control, helping to strike a balance between freely admitting the CO2 needed for photosynthesis and at the same time preventing excessive loss of water vapor from the plant. Air pollutants such as ozone (O3), nitrous oxide (NO), and sulfur dioxide (SO2) also enter plants primarily through the open stomata.

1.1C. Vascular Tissue

The xylem and the phloem make up the vascular systems found contiguously in the roots, stems (Fig. 1-3), and leaves of plants. In a tree trunk, the phloem constitutes a layer of the bark and the xylem constitutes almost all of the wood. The xylem provides structural support for land plants. Water conduction in the xylem of a tree often occurs only in the outermost annual ring,¹ which lies just inside the vascular cambium (region of meristematic activity from which xylem and phloem cells differentiate). Outside the functioning phloem 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 and into 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 found abaxial 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 and rebranches as it crosses a typical dicotyledonous leaf, becoming smaller (in cross section) at each step.² In contrast to the reticulate venation in dicotyledons, monocotyledons characteristically have parallel-veined leaves, a readily observable characteristic. Individual mesophyll cells in a leaf are usually within a few cells of the vascular tissue.

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.

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 about 10  mol  m−³ (10  mM)³ inorganic nutrients plus organic forms of nitrogen that are metabolically produced in the root. The xylem is a tissue of various cell types that we will consider in more detail in the final chapter (Section 9.4B,D), when water movement in plants is discussed quantitatively. The conducting cells in the xylem are the narrow, elongated tracheids and the vessel members (also called vessel elements), which tend to be shorter and wider than the tracheids. Vessel members are joined end to end in long linear files; their adjoining end walls or perforation plates have from one large hole to many small holes. The conducting cells lose their protoplasts, and the remaining cell walls thus form a low-resistance channel for the passage of solutions. Xylem sap moves from the root, up the stem, through the petiole, and then to the leaves in these hollow dead xylem cells, with motion occurring in the direction of decreasing hydrostatic pressure. Some solutes 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 cells adjacent to the conducting cells of the xylem.

The movement of most organic compounds throughout the plant takes place in the other vascular tissue, the phloem. A portion of the photosynthetic products made in the mesophyll cells of the leaf diffuses or is actively transported 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 throughout the plant. The carbohydrates produced by photosynthesis and certain other substances generally move in the phloem toward regions of lower concentration, although diffusion is not the mechanism for the movement, as indicated in Chapter 9 (Section 9.4F,G). The phloem is a tissue consisting of several types of cells. In contrast to the xylem, however, the conducting cells of the phloem contain cytoplasm. They are known as sieve cells and sieve tube members (Fig. 1-3) and are joined end to end, thus forming a transport system throughout the plant. Although these phloem cells often contain no nuclei 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 nutrients from the soil and then conduct these substances inward and then upward to the stem. Approximately half of the products of photosynthesis are allocated to roots for many plants. To help understand uptake of substances into a plant, we will examine the cell types and the functional zones that occur along the length of a root.

At the extreme tip of a root is the root cap (Fig. 1-4a), which consists of relatively undifferentiated cells that are scraped off as the root grows into new 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 a meristematic region where the cells rapidly divide. Cells in this apical meristem tend to be isodiametric and have thin cell walls. Next is a region of cell elongation in the direction of the root axis. Such elongation mechanically pushes 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 spatially distinct from the elongation zone. Also, cell size and the extent of the zones vary with both plant species and physiological status.

The next region indicated in Figure 1-4a is that of cell differentiation, where the cells begin to assume more highly specialized functions. The cell walls become thicker, and elongation is greatly diminished. The epidermal cells develop fine projections, radially outward from the root, called root hairs. These root hairs greatly increase the surface area across which water and nutrients can enter a plant. As we follow a root toward the stem, the root surface generally becomes less permeable to water and the root interior becomes more involved with conducting water toward the stem. Water movement into the root is discussed in Chapter 9 (Section 9.4A), so the discussion here is restricted to some of the morphological features.

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, indicating the arrangement of the various cell types.

The region of a young root where water absorption most readily occurs usually has little or no waxy cuticle. Figure 1-4b shows a cross section of a root at the level where root hairs are found. Starting from the outside, we observe first the root epidermis and then a number of layers of cells known as the cortex. Abundant intercellular air spaces occur in the cortex, facilitating the diffusion of O2 and CO2 within this tissue (such air spaces generally are lacking in vascular tissue). Inside the cortex is a single layer of cells, the endodermis. The radial and transverse walls of the endodermal cells are impregnated with waxy material, including suberin, forming a band around the cells known as the Casparian strip (Fig. 1-4), which prevents passage of water and solutes across that part of the cell wall. Because there are no air spaces between endodermal cells, and the radial walls are blocked by the waterproof Casparian strip, water must pass through the lateral walls and enter the cytoplasm of endodermal cells to continue across a root. The endodermal cells can represent the only place in the entire pathway for water movement from the soil, through the plant, to the air where it is mandatory that the water enters a cell's cytoplasm.⁴ In the rest of the pathway, water can move in cell walls or in the hollow lumens of xylem vessels, a region referred to as the apoplast.

Immediately inside the endodermis is the pericycle, which is typically one cell thick in angiosperms. The cells of the pericycle can divide and form a meristematic region that can produce lateral or branch roots in the region just above the root hairs. Radially inside the pericycle is the vascular tissue. The phloem generally occurs in two to eight or more strands located around the root axis. The xylem usually radiates out between the phloem strands, so water 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; through cell division and differentiation, it produces xylem (to the inside in stems and older roots) and phloem (to the outside in stems and older roots).

Our rather elementary discussion of leaves, vascular tissues, and roots leads to the following oversimplified but useful picture. The roots take up water from the soil along with nutrients required for growth. These are conducted in the xylem to the leaves. Leaves of a photosynthesizing plant lose the water to the atmosphere along with a release of O2 and an uptake of CO2. Carbon from the latter ends up in photosynthate translocated in the 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 functionally interacting with the environment. To understand the details of such physiological processes, we must turn to fields like calculus, physics, thermodynamics, and photochemistry. Our next step is to bring the abstract ideas of these fields into the realm of cells and plants, which means that we need to make 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 thermodynamic concept of chemical potential. This leads to a quantitative description of fluxes, electrical potentials across membranes, and the energy requirements for active transport of solutes. Some important energy conversion processes take 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 energy useful 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 environment through exchanges of water vapor and CO2. In our problem-solving approach to these topics, we will pay particular attention to dimensions and ranges for the parameters as well as to the insights that can be gained by developing the relevant formulas and then making calculations.

1.2. Diffusion

Diffusion leads to the net movement of a substance from a region of higher concentration 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.

Diffusion takes place in both the liquid and the gas phases associated with plants and is a result of the random thermal motion of the molecules—the solute(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 greater probability exists for molecules to move from the concentrated region to the dilute region than vice versa (Fig. 1-5). In other words, more molecules 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 available for movement in the opposite direction. If left isolated from external influences, diffusion of a neutral species tends to even out concentration differences originally present in adjoining regions of a liquid or a gas. In fact, the randomizing tendency of the generally small, irregular motion of particles by diffusion is a good example of the increase in entropy or disorder that accompanies all spontaneous processes. In 1905, Albert Einstein, the German-born theoretical physicist and later American citizen (who received the Nobel Prize in Physics in 1921), described such diffusion as a case of Brownian motion or movement, which was first observed microscopically by the Scottish botanist Robert Brown in 1827 for colloidal particles.

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 is involved in many plant processes, such as gas exchange and the movement of nutrients toward root surfaces. For instance, diffusion is the mechanism for most, if not all, steps by which CO2 from the air reaches the sites of photosynthesis in chloroplasts. CO2 diffuses from the atmosphere up to the leaf surface and then diffuses through the stomatal pores. After entering a leaf, CO2 diffuses within intercellular air spaces (Fig. 1-2). Next, CO2 diffuses across the cell wall, crosses the plasma membrane of a leaf mesophyll cell, and then diffuses through the cytosol to reach the chloroplasts (Fig. 1-1). Finally, CO2 enters a chloroplast and diffuses up to the enzymes that are involved in carbohydrate formation. If the enzymes were to fix all of the CO2 in their vicinity, and no other CO2 were to diffuse in from the atmosphere surrounding the plant, photosynthetic processes would stop (in solution, CO2 can also occur in the form of bicarbonate, HCO3–, and the crossing of membranes does not have to be by diffusion, refinements that we will return to in Chapter 8, Section 8.3D). In this chapter we develop the mathematical formulation necessary for understanding both diffusion across a membrane and diffusion in a solution.

1.2A. Fick's First Law

In 1855, the German physiologist Adolph Eugen Fick was one of the first to examine diffusion quantitatively. For such an analysis, we need to consider the concentration (cj) of some solute species j in a solution or gaseous species j in air; the subscript j indicates 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 diffusion from the concentrated to the dilute region (Fig. 1-5; strictly speaking, this applies to neutral molecules or in the absence of electrical potential differences, an aspect that we will return to in Chapter 3, Section 3.2). Such a molecular flow down a concentration gradient is analogous to the flow of heat from a warmer to a cooler region. The analogy is actually good (especially for gases) because both processes depend on the random thermal motion of molecules. In fact, the differential equations and their solutions that are used to describe diffusion are those that had previously been developed to describe heat flow.

To express diffusion quantitatively, we will consider a diffusive flux or flow of species j. For simplicity, we will restrict our attention to diffusion involving planar fronts of uniform concentration, a situation that has widespread application to situations of interest in biology. We will let Jj be the amount of species j crossing a certain area per unit time, for example, moles of particles per meter squared in a second, which is termed the flux density.⁵ Reasoning by analogy with heat flow, Fick deduced that the force, or causative agent, leading to the net molecular movement involves the concentration gradient. A gradient indicates how a certain parameter changes with distance; the gradient in concentration of species j in the x-direction is represented by ∂cj/∂x. This partial derivative, ∂cj/∂x, indicates how much cj 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, the flux density of some substance is proportional to an appropriate force, a relation 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, which we will represent by –∂cj/∂x for diffusion in one dimension. To help appreciate why a negative sign occurs, recall that the direction of net (positive) diffusion is toward regions of lower concentration. We can now write the following relation showing the dependence of the flux density on the driving force:

(1.1)

where Dj is used to transform the proportionality between flux density and the negative concentration gradient into an equality.

Equation 1.1 is commonly known as Fick's first law of diffusion, where Dj is the diffusion coefficient of species j. For Jj in mol m−² s−¹ and cj in mol m−³ (hence, ∂cj/∂x in mol m−⁴), Dj has units of m² s−¹. Because Dj varies with concentration, temperature, and the medium for diffusion, it is properly called a coefficient in the general case. In certain applications, however, we can obtain sufficient accuracy by treating Dj as a constant. The partial derivative is used in Equation 1.1 to indicate the change in concentration in the x-direction of Cartesian coordinates 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 the x-direction has the same magnitude at any value of y and z, meaning that we are dealing with one-dimensional, planar fluxes. By convention, a net flow in the direction of increasing x is positive (from left to right in Fig. 1-6). Because a net flow occurs toward regions of lower concentration, we again note that the negative sign is needed in Equation 1.1. Fick's first law indicates that diffusion is greater when the concentration gradient is steeper (higher) or the diffusion constant is larger. It has been amply demonstrated experimentally and is the starting point for our formal discussion of diffusion.

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 ).

1.2B. Continuity Equation and Fick's Second Law

As we indicated earlier, diffusion in a solution is important for the movement of solutes across plant cells and tissues. How rapid are such processes? For example, if we release a certain amount of material in one location, how long will it take before we can detect that substance at various distances? To discuss such phenomena adequately, we must determine the dependence of the concentration on both time and distance. We can readily derive such a time–distance relationship if we first consider the conservation of mass, which is necessary if we are to transform Equation 1.1 into an expression that is convenient for describing the actual solute distributions caused by diffusion. In particular, we want to eliminate Jj from Equation 1.1 so that we can see how cj depends on x and t.

The amount of solute or gaseous species j per unit time crossing a given area, here considered to be a planar area perpendicular to the x-axis (Fig. 1-6), can change with position along the x-axis. Let us imagine a volume element of thickness dx in the direction of flow and of cross-sectional area A (Fig. 1-6). At x, we will let the flux density across the surface of area A be Jj. At x  +  dx, the flux density has changed to Jj + (∂Jj/∂x)dx, where ∂Jj/∂x is the gradient of the flux density of species j in the x-direction; that is, the rate of change of Jj with position, ∂Jj/∂x, multiplied by the distance, dx, gives the overall change in the flux density, or dJj = (∂Jj/∂x) dx.

The change in the amount of species j in the volume Adx in unit time for this one-dimensional case is the amount flowing into the volume element per unit time, JjA, minus that flowing out, [Jj  +  (∂Jj/∂x)dx]A. The concentration of species j (cj) is the amount of 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 the concentration of species j with time, ∂cj/∂t, multiplied by the volume in which the change in concentration occurs, Adx. Equating these two different expressions that describe the rate of change in the amount of species j in the volume Adx, we obtain the following relation:

(1.2)

The two JjA terms on the left-hand side of Equation 1.2 cancel each other. Then after division through by Adx, Equation 1.2 leads to the very useful expression known as the continuity equation:

(1.3)

The continuity equation is a mathematical way of stating that matter cannot be created or destroyed under ordinary conditions. Thus, if the flux density of some species decreases as we move in the x-direction (∂Jj/∂x  <  0), Equation 1.3 indicates that its concentration must be increasing with time, as the 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 the important special case of constant Dj, this general equation for diffusion becomes

(1.4)

The solution of Equation 1.4, which is the most difficult differential equation to be encountered in this book (Crank, 1999; Lamb, 2019), describes how the concentration of some species changes with position and time as a result of diffusion. To determine the particular function that satisfies this important differential equation, we need to know the specific conditions for the situation under consideration. Nevertheless, a representative solution useful for the consideration of diffusion under simple conditions will be sufficient for the present purpose of describing the characteristics of solute diffusion in general terms. For example, we will assume that no obstructions occur in the x-direction and that species j is initially placed in a plane at the origin (x  =  0). In this case, the following expression for the concentration of species j satisfies the differential form of Fick's second law when Dj is constant⁶:

(1.5)

In Equation 1.5, Mj is the total amount of species j per unit area initially (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 here), and cj is its concentration at position x at any later time t. For Mj , where cj(x, t). Moreover, the solute can be allowed to diffuse for an unlimited distance in either the plus or the minus x-direction and no additional solute is added at times t  >  0. Often this idealized situation can be achieved by inserting a radioactive tracer in a plane at the 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 second law. The form is relatively simple compared with other solutions, and, more important, the condition of having a finite amount of material released at a particular location is realistic for certain applications to biological problems.

1.2C. Time–Distance Relation for Diffusion

Although the functional form of cj given by Equation 1.5 is only one particular solution to Fick's second law (Eq. 1.4) and is restricted to the case of constant Dj, it nevertheless is an extremely useful expression for understanding diffusion. It relates the distance a substance diffuses to the time necessary to reach that distance. The expression uses the diffusion coefficient of species j, Dj, which can be determined experimentally. In fact, Equation 1.5 is often used to determine a particular Dj.

Equation 1.5 indicates that the concentration in the plane at the origin of the x-direction (x , which becomes infinitely large as t is turned back to 0, the starting time. Practically speaking, this extrapolated infinite value for cj at x equals 0 corresponds to having all of the solute initially placed as close as possible to a plane at the origin. For t greater than 0, the solute diffuses away from the origin. The distribution of molecules along the x-axis at two successive times is indicated in Figures 1-7a and 1-7b, whereas Figures 1-7c and 1-7d explicitly show the movement of the concentration profiles along the time axis. Because the total amount of species j does not change (it remains at Mj per unit area of the y–z plane, 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. Comparing Figures 1-7a and 1-7b, we can see that the average distance of the diffusing molecules from the origin increases with time. Also, Figures 1-7c and 1-7d show how the concentration profiles flatten out as time increases, as the diffusing solute or gaseous species is then distributed over a greater region of space.

In estimating how far molecules diffuse in time t, a useful parameter is the distance x1/e at which the concentration drops to 1/e or 37% of its value in the plane at the origin. Although somewhat arbitrary, this parameter describes the shift of the statistical distribution of the population of molecules with time. From (Fig. 1-7). The concentration therefore drops to 1/e (= e−¹) of the value at the origin when the exponent of e , so the distance x1/e satisfies

(1.6)

The distance x1/e along the x-axis is indicated in Figures 1-7a, 1-7b, and 1-7d.

Figure 1-7  Concentration of species j, c j , 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 , c j 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 t a , (b) distribution occurring at a subsequent time t b , (c) portrayal of the concentration profiles at t a and t b , and (d) three-dimensional surface portraying change of concentration with time and position. Note that x 1/ e is the location at which the concentration of species j has dropped to 1/ e of its value at the origin.

Equation 1.6 is an extremely important relationship that indicates a fundamental characteristic of diffusion processes: The distance a population of molecules of a particular solute or gaseous species diffuses—for the one-dimensional case in which the molecules are released in a plane at the origin—is proportional to the square root of both the diffusion coefficient of that species and the time for diffusion. In other words, the time to diffuse a given distance increases with the square of that distance. An individual molecule may diffuse a greater or lesser distance in time t1/e than is indicated by Equation 1.6 (Fig. 1-7) because the latter refers to the time required for the concentration of species j at position x1/e to become 1/e of the value at the origin; that is, we are dealing with the characteristics of a whole population of molecules, not the details of an individual molecule. Furthermore, the factor 4 is rather arbitrary because some criterion other than 1/e causes this numerical factor to be somewhat different, although the basic form of Equation 1.6 is preserved. For example, the numerical factor is (ln 2)(4) or 2.8 if the criterion is to drop to half of the value at the origin.

Table 1-1 lists the magnitudes of diffusion coefficients for various solutes in water at 25°C.⁷ For ions and other small molecules, Dj's in aqueous solutions are approximately 10−⁹  m²  s−¹. Because proteins have higher relative molecular masses (i.e., higher molecular weights)⁸ than the small solutes, their diffusion coefficients are lower (Table 1-1). Also, because of the greater frictional interaction between water molecules and fibrous proteins than with the more compact globular ones, fibrous proteins often have diffusion coefficients that are approximately half of those of globular proteins of the same molecular weight.

Table 1-1

a Values are for dilute solutions at 25°C or air under standard atmospheric pressure at 20°C (sources: Lundblad and MacDonald, 2010; Rumble, 2018).

To illustrate the time–distance consequences of Equation 1.6, we quantitatively consider the diffusion of small molecules in an aqueous solution. How long, on average, does it take for a small solute with a Dj of 1  ×  10−⁹  m²  s−¹ to diffuse 50  μm, the distance across a typical leaf cell? From Equation 1.6, the time required for the population of molecules to shift so that the concentration at this distance is 1/e of the value at the origin is

Thus, diffusion is fairly rapid over subcellular distances.

Next, let us consider the diffusion of the same substance over a distance of 1  m. The time needed is

Diffusion is indeed not rapid over long distances! Therefore, inorganic nutrients in xylary sap do not ascend a tree by diffusion at a rate sufficient to sustain growth. On the other hand, diffusion is often adequate for the movement of solutes within leaf cells and especially inside organelles such as chloroplasts and mitochondria. In summary, diffusion in a solution is fairly rapid over short distances (less than about 100  μm) but extremely slow for very long distances.

In living cells, cytoplasmic streaming (cyclosis) causes mechanical mixing, which leads to much more rapid movement than by diffusion. This cytoplasmic streaming, whose cessation is often a good indicator that cellular damage has occurred, requires energy, which is usually supplied in the form of adenosine triphosphate (ATP). The movement can involve actin microfilaments and microtubules (Duan and Tominaga, 2018; Iwabuchi et al., 2019). Chloroplasts can also move around in some cells in response to changes in illumination.

1.2D. Diffusion in Air

Diffusion of gases in the air surrounding and within leaves is necessary for both photosynthesis and transpiration. For instance, water vapor evaporating from the cell walls of mesophyll cells diffuses across the intercellular air spaces (Fig. 1-2) to reach the stomata and from there diffuses across an air boundary layer into the atmosphere (considered in detail in Chapter 8, Section 8.2). CO2 diffuses from the atmosphere through the open stomata to the surfaces of mesophyll cells, and the photosynthetically evolved O2 traverses the same pathway in the reverse direction, also by diffusion. The experimentally determined diffusion coefficients of these three gases in air at sea level (standard atmospheric pressure) and 20°C are about 2  ×  10−⁵  m²  s−¹ (Table 1-1). Such diffusion coefficients in air are approximately 10⁴ times larger than the Dj describing diffusion of a small solute in a liquid, indicating that diffusion coefficients depend markedly on the medium. In particular, many more intermolecular collisions occur per unit time in a liquid phase than in a less dense gas phase. Thus, a molecule can move further and faster in air than in an aqueous solution before being influenced by other molecules.

in a gas phase facilitate diffusion. Spongy tissue known as aerenchyma with particularly large intercellular air spaces and even air channels can develop in the cortex of stems and roots and in leaves; again, advantage is taken of the approximately 10,000-fold larger diffusion coefficients of gases in air compared with in water. Indeed, aerenchyma is crucial for plants in aquatic, wetland, and flood-prone habitats, e.g., water lilies. This low-resistance pathway of air-filled channels facilitates gas exchange between plant organs above and below the local water level (insects also rely on internal air pathways to move O2 and CO2 around).

Although the relation between diffusion