Cell membrane

The cell membrane, also called the plasma membrane, is the selective boundary separating a biological cell's cytoplasm from its surroundings. Both prokaryotes and eukaryotes have a cell membrane. Where a cell wall is present, it is a distinct structure outside the cell membrane; the two names are not synonyms. This separation lets a biological cell maintain internal conditions while exchanging material through membrane transport. See NHGRI: cell membrane.

The familiar architecture is a lipid bilayer containing membrane proteins. In a phospholipid-based lipid bilayer, two membrane leaflets face opposite ways: hydrophilic heads face the water on each side, while hydrophobic tails face one another in the interior. The heads are therefore exposed both outside and inside the biological cell. This arrangement combines a water-facing surface with a barrier to many water-soluble solutes; the schematic represents organization rather than an atomistic structure. See Alberts et al., Molecular Biology of the Cell: the lipid bilayer.

Selective permeability means that different substances cross at different rates under comparable conditions. Small nonpolar molecules can pass through a lipid bilayer by diffusion, while ions encounter a strong barrier without a suitable route. An ion channel supplies a selective water-filled pathway when open; a carrier protein binds a solute and changes shape to expose it to the other side. These membrane proteins therefore add specific routes to the barrier. An open ion channel permits passive transport, whereas a carrier protein can mediate either passive transport or active transport. See Alberts et al.: principles of membrane transport.

The useful combination is retention plus controlled exchange: a completely impermeable boundary would prevent uptake, whereas unrestricted mixing would erase differences in concentration and membrane potential. The fluid mosaic model connects these functions to an organized but mobile lipid bilayer containing proteins.

GLUT1 is a facilitative glucose transporter expressed in many tissues. Human beta cells can rely predominantly on GLUT1 rather than sharing the high GLUT2 abundance of rodent beta cells.

GLUT2 is a facilitative glucose transporter with important roles in hepatocytes and in the classical rodent beta-cell model. It supports transport down a concentration gradient and should not be assumed to be the predominant human beta-cell glucose transporter.

The lipid divide is the contrast between characteristic archaeal membrane lipids and characteristic bacterial/eukaryotic membrane lipids, including glycerol stereochemistry and ether versus ester linkages. It constrains hypotheses for eukaryotic origins without uniquely determining the sequence of ancestral membrane changes.

Membrane fission separates a connected membrane into distinct components, for example by severing a vesicle neck. Bending a membrane into a bud without severing that neck is not yet fission.

Membrane fusion joins initially separate lipid bilayers, changing connections between membrane compartments. It requires controlled rearrangement of both leaflets while avoiding uncontrolled leakage.

A measurement of exchange between previously separate enclosed aqueous compartments. Leakage into the surrounding medium is a different event and needs a separate control.

A readout of lipid redistribution between membrane populations, for example through fluorescence dequenching. It must be distinguished from aqueous content mixing and leakage.

Membrane fusion intermediate in which contacting membrane leaflets have merged while the aqueous compartments remain separate. A lipid mixing assay therefore need not establish full membrane fusion.

A trans-SNARE complex links SNARE proteins anchored in different lipid bilayers. Its formation can bring membranes together before complete membrane fusion.

Membrane shape affects the energetic cost and geometry of a bud, neck or protrusion. Proteins, lipid composition and mechanical forces can change curvature; curvature alone does not perform the topological reconnection required for membrane fusion or membrane fission.

homeoviscous adaptation changes membrane composition in response to conditions such as growth temperature, helping preserve useful membrane dynamics. The original Escherichia coli study measured temperature-dependent fatty-acid changes and membrane-lipid viscosity by spin labeling.^[ref]

A membrane nanodomain is a small region with locally differing composition or molecular organization. Its size, lifetime and physical origin require measurement; detergent resistance alone does not demonstrate a pre-existing domain in a living membrane.

Selective permeability means that a cell membrane permits different substances to pass at different rates under comparable conditions. Both the lipid bilayer and the available membrane proteins contribute to this selection.

Artificial boundaries can also have selective permeability. For osmosis, specify which solutes cross and over what observation interval: a boundary that retains one solute need not retain another. A semipermeable membrane is the ideal solvent-only boundary used in the osmotic pressure calculation.

membrane permeability describes passage of a specified substance through a membrane under stated conditions. It is distinct from lateral diffusion within the membrane and from the mechanical force needed to rupture the membrane.

A semipermeable membrane ideally passes the solvent while excluding the specified solutes. In the aqueous osmosis model below, water can cross but the counted dissolved particles cannot. Real selective permeability is often a matter of different rates, so this idealization needs checking over the measurement interval. Neither an impermeable sheet nor a freely mixing opening realizes this model.^[ref]

A membrane protein is a protein associated with a biological membrane. Its association, orientation and mobility are separate properties: being embedded does not imply spanning, and spanning does not imply moving freely. Membrane proteins supply routes for membrane transport and recognition for signal transduction.^[ref]

An ion channel supplies a pathway for ions; a carrier protein changes which side can reach a bound solute; a cell-surface receptor connects an external ligand to intracellular signal transduction. These examples distinguish passage from communication: a signal can cross the boundary as a change of protein activity without the ligand itself crossing.^[ref]

The same boundary can therefore retain substances and communicate selectively with its surroundings. Membrane topology tells us which parts can meet a given ligand; lateral mobility of membrane proteins tells us whether two parts can approach each other along the boundary. A diagram of a cell membrane becomes more useful when it answers both questions.

A light-driven proton pump with a seven-helix membrane-spanning structure and a retinal chromophore. Its mechanical unfolding involves membrane anchoring as well as protein and lipid interactions.

An iron-export protein. IRE-containing transcripts can be regulated at biological translation, while protein-level and systemic regulation provide additional controls.

A receptor supporting uptake of transferrin-bound iron. RNA regulation involving its 3′ region can increase transcript stability during iron shortage.

A bacterial outer-membrane protein studied by NMR in optimized lipid nanodiscs.^[ref]

An experimental environment designed to shield hydrophobic protein surfaces while retaining relevant membrane properties. Detergent micelles, bicelles and lipid bilayer nanodiscs have different limitations.

A cell-surface protein whose phosphorylated intracellular region can recruit signalling adaptors.

A cargo receptor links a transported component to sorting machinery. An ER export receptor can bind a soluble lumenal cargo while exposing a cytoplasmic recognition site to the coat.

An integrin is a transmembrane adhesion receptor linking extracellular ligands to intracellular signalling and cytoskeleton interactions. Integrin-dependent contacts transmit force in many forms of cell migration; their necessity depends on the physical environment.

hydrophobic mismatch is an energetic incompatibility between a membrane protein's hydrophobic span and the hydrophobic thickness of its surrounding lipid bilayer. Local deformation, altered conformation or changed association can partly accommodate it.

An integral membrane protein is firmly incorporated into a membrane, commonly through contact with the hydrophobic interior of the lipid bilayer. Extraction behavior is an operational distinction: it resists treatments that remove many peripheral membrane proteins. Transmembrane proteins are one class; integral does not mean spanning.^[ref]

A transmembrane protein has at least one polypeptide chain crossing the complete lipid bilayer. One crossing is single-pass; several crossings in one chain are multipass. Hydrophobic regions face the membrane interior, while hydrophilic regions can face water.^[ref]

A tail-anchored protein has a membrane-spanning segment near its C-terminus; many reach the endoplasmic reticulum after biological translation.

A peripheral membrane protein associates with a membrane face through noncovalent interactions, without spanning its hydrophobic interior. Association with another membrane protein can suffice; peripheral describes attachment, not an unimportant function.^[ref]

A lipid-anchored protein attaches through a covalently linked lipid. The anchor inserts into the lipid bilayer; the polypeptide chain need not cross it. This is a description of attachment rather than a synonym for transmembrane protein.^[ref]

Membrane topology specifies the sidedness of a membrane-associated polypeptide chain: where its N-terminus, C-terminus and intervening protein domains lie relative to the lipid bilayer. Side-specific placement is essential to interpreting a cell-surface receptor.^[ref]

For an original worked example, stipulate a continuous polypeptide chain with its N-terminus outside a biological cell, exactly three complete crossings of the cell membrane, and no additional excursions into the lipid bilayer. Its segments lie, in order, outside, inside, outside, inside. A protein domain between the first and second crossings faces the cytosol; one between the second and third faces the exterior; the C-terminus faces the cytosol. Counting crossings determines these placements without specifying an atomic protein structure.

The proof is a side-switching argument. Assign exterior the sign $+1$ and interior $-1$. Each complete crossing multiplies the sign by $-1$, so after $m$ crossings,An odd number puts the termini on opposite sides, an even number on the same side. This assumes complete crossings of one boundary; a chain that enters and returns to the same side without crossing must not be counted as a complete crossing. Nor does the rule determine which side the N-terminus starts on.

As a constructed accessibility test, put an epitope on the outside segment between crossings two and three, and another on the inside segment between crossings one and two. Stipulate an intact cell membrane that excludes the test antibody and accessible epitopes that bind it. Applying the antibody outside can label only the first epitope. If both become accessible after opening the boundary, the difference tests sidedness under those assumptions. Failure to label an epitope alone is insufficient: it could also be inaccessible because of protein folding or poor antibody affinity. This is a proposed reasoning exercise, separate from the historical mixing experiment below.

Lateral mobility of membrane proteins is movement along a membrane surface. It differs from membrane transport through the boundary and from reversing membrane topology. Attachment, crowding and barriers can restrict diffusion; the fluid mosaic model does not require every membrane protein to wander freely.^[ref]

An original local model treats a small, approximately flat patch as an unbounded plane with uniform diffusion coefficient $D$, no drift and no trapping. Represent the two displacement coordinates by independent Brownian motions, each with zero expected value and variance $2Dt$. Since $r^2=x^2+y^2$, addition of the two coordinate expectations givesThis is a root-mean-square displacement, not distance traveled along a jagged trajectory or the displacement of every protein. For a stipulated $D=0.08\mu {\mathrm{m}}^2{\mathrm{s}}^{-1}$ and $t=5\mathrm{s}$, $⟨r^2⟩=1.6\mu {\mathrm{m}}^2$ and $r_{\mathrm{rms}}\approx 1.265\mu \mathrm{m}$. Reaching a stipulated root-mean-square scale $L=2\mu \mathrm{m}$ gives $t=L^2/(4D)=12.5\mathrm{s}$. These model values are not measured diffusion coefficients or mixing times from the historical experiment.

Multiplying the target displacement scale by two multiplies the model time by four. This quadratic scaling is a useful way to see why local encounters and redistribution over an entire biological cell need not happen on the same timescale. A curved, bounded cell membrane, restricted paths and different mobile populations require more than this flat-patch model; disappearance of an initial pattern alone does not determine $D$.

In 1970 Larry Frye and Michael Edidin fused mouse C11D and human VA-2 cell lines with Sendai virus, forming heterokaryons. Suspensions were maintained in a shaking water bath at $37{}^{\circ }\mathrm{C}$. Removed aliquots were stained by indirect immunofluorescence: antibodies against parental surface antigens, then differently fluorescent secondary antibodies. A Leitz Ortholux fluorescence microscope used a mercury lamp and color-selecting filters. Green mouse and red human surface patterns became intermixed.^[ref]

Same-species fusions tested antibody specificity; stable hybrid lines supplied a two-color control. Inhibitors of biological translation did not prevent mixing; lowering temperature inhibited it. These observations supported lateral redistribution over rapid synthesis. The authors could not conclusively exclude passage through cytoplasm and return. The readout compared populations, not trajectories of a continuously observed individual biological cell, and did not establish unrestricted motion of every membrane protein.^[ref]

The decisive design idea is to begin with spatially distinguishable material. A uniformly labeled surface can look unchanged while its components move. Bringing together differently identifiable parental surfaces creates an initial pattern whose loss reveals redistribution; antibody specificity makes that loss interpretable. Membrane topology asks which side a part faces, whereas this experiment asks where it can move along that side.

A nucleotide-sugar transporter supplies an activated donor across a membrane to a compartment containing glycosyltransferases. Whole-cell donor abundance does not establish donor availability at a lumen-facing enzyme.

A scramblase facilitates relatively nonselective bidirectional movement of lipids between membrane leaflets, tending to dissipate asymmetry when active. Its action differs from directional transport used to maintain asymmetry.

A flippase catalyzes directional movement of selected lipids between membrane leaflets. Such activity can establish or maintain lipid asymmetry, subject to specificity and energy coupling.

An ATP-binding cassette transporter couples nucleotide-binding-domain activity to transmembrane transport. For an exporter, productive substrate movement must be measured separately from substrate binding and ATP hydrolysis.

AcrB is a bacterial proton-coupled multidrug transporter that functions as part of the AcrAB–TolC export assembly. Its asymmetric trimer and multiple drug-binding regions are structural models for studying broad specificity.^[ref]

A multidrug efflux pump exports multiple chemical substrates using a coupled source of free energy. Broad recognition can arise from overlapping binding sites and conformational flexibility, but binding alone does not establish productive transport.

A membrane carrier protein binds a solute and changes its exposed binding site from one side of a cell membrane to the other. Depending on its mechanism and coupling, it can mediate passive transport or active transport.

An ion channel is a membrane protein that provides a selective pathway for ions across a cell membrane. An open pore permits passive transport down the relevant electrochemical gradient; it does not supply the free energy for uphill transport as a pump involved in active transport does. Two things are needed for a net electric current: an available pathway and an electrochemical potential difference. Opening a pore alone need not produce a net electric current. See Ion Channels and the Electrical Properties of Membranes.

The selective permeability of an ion channel depends on interactions between its pore and the passing ions, not simply on a hole's diameter. Some ion channels strongly favor potassium over sodium; others admit several kinds of positively charged ions. Thus selective does not necessarily mean exclusive to one ion species. See Purves et al., Neuroscience, 2nd edition (2001): molecular structure of ion channels and the mixed-channel example in Purves et al.: principles derived from studies of the neuromuscular junction.

Gating is the opening and closing of an ion channel's pathway. A voltage-gated ion channel responds to membrane potential; an ionotropic receptor can respond to a bound ligand. Gating changes the available electrical conductance, while the electrochemical gradient supplies the drive for passive transport. For a fixed reversal potential in the simple model below, opening more identical ion channels increases the magnitude of the electric current at a fixed nonzero drive, without choosing its sign independently of membrane potential. The two-state ion-channel gating model describes how the open fraction can change. See the Alberts reference above.

For a worked approximation, hold an open population's electrical conductance at $g=10\mathrm{nS}$ and its reversal potential at $E_{\mathrm{rev}}=-80\mathrm{mV}$. Use $V=V_{\mathrm{inside}}-V_{\mathrm{outside}}$ for membrane potential and inward-positive conventional current:At $V=-100,-80,-60\mathrm{mV}$, respectively, the electric currents are $+200,0,-200\mathrm{pA}$, since $1\mathrm{nS}\times 1\mathrm{mV}=1\mathrm{pA}$. Positive electric current means net positive electric charge entering, or net negative electric charge leaving; it does not mean that every ion moves inward. For a pore passing only positive ions, the three results mean net inward passage, no net passage, and net outward passage. These stipulated values describe an illustrative Ohmic approximation, not measured properties of a particular ion channel; see reversal potential for the current plot and limitations.

The useful connection is between a molecular gate and a controllable electric current. A ligand can change electrical conductance through an ionotropic receptor, or membrane potential can change it through a voltage-gated ion channel, connecting recognition and electrical signaling. In a historical landmark, Erwin Neher and Bert Sakmann reported currents from individual ion channels in frog muscle membranes in 1976. See Neher and Sakmann, Single-channel currents recorded from membrane of denervated frog muscle fibres, Nature 260, 799–802 (1976).

The probability that an ion channel is in a conducting open state under specified conditions. For N equivalent channels, each carrying open-state electric current i, the mean total electric current is $N{P}_{O}i$. An ensemble open fraction and the fraction of time one channel remains open coincide under suitable stationary and ergodic assumptions. A two-state ion-channel gating model is one model of this probability, not its definition.

For N independent equivalent ion channels with open probability $P_O$ and approximately Ohmic open-state electrical conductance $\gamma$, the mean electric current is $I=N{P}_O\gamma (V-E)$. A numerical Electrical conductance cannot be inferred without the relevant reversal potential or driving voltage.

An ion channel that conducts chloride and, for some family members, other anions. Members differ in voltage dependence and physiological role; an anion channel is distinct from a coupled anion exchanger.

A guard-cell outward-rectifying potassium channel. Membrane depolarization favors potassium efflux through this channel during stomatal closure.

A slow anion channel important for guard-cell anion efflux. Regulatory phosphorylation links ABA-responsive and calcium-dependent kinases to channel activation.

A mechanosensitive ion channel changes gating in response to mechanical conditions transmitted through the membrane or associated structures. A changed calcium signal alone does not identify which channel or mechanical coupling was responsible.

A large-conductance bacterial mechanosensitive channel serving as an emergency release pathway under high membrane tension.

A bacterial mechanosensitive channel that can release solutes during sudden osmotic downshock.

Channel desensitization is entry into a less responsive or nonconducting state during sustained stimulation. A ligand-bound channel can therefore be closed without the ligand having dissociated.

A ligand-gated ion channel couples ligand occupancy to the probability of pore opening. The ligand-binding site and ion pathway can be distinct, and sustained occupancy can lead to a desensitized nonconducting state.

A selectivity filter is a channel-pore region that preferentially stabilizes permeating ions through geometry and chemical interactions. Selectivity balances dehydration and coordination energies rather than following pore diameter alone.

Membrane transport is the passage of substances across a cell membrane. For the small solutes considered here, distinguish the available route from its driving force: a lipid bilayer, ion channel, or carrier protein determines how passage can occur, while differences in chemical potential and membrane potential determine which direction is energetically downhill. A large driving force cannot make an unavailable route conduct. See Alberts et al.: principles of membrane transport.

For an ideal dilute chemical solution at uniform temperature and pressure, with the same reference molar chemical potential on both sides, the molar electrochemical potential change for transferring a solute from compartment A to B isHere $c_{\mathrm{A}},c_{\mathrm{B}}$ are concentrations, $T$ is absolute temperature, $R$ is the molar gas constant, $z$ is signed ion charge number, $F$ is the Faraday constant, and $V_{\mathrm{B}}-V_{\mathrm{A}}$ is the electric potential difference. The natural logarithm contains a dimensionless ratio. The electrical term vanishes for an uncharged solute. More generally, use thermodynamic activities in place of the ideal concentration ratio. This combines the ideal-solution chemical potential difference with electrical energy per mole; see IUPAC Gold Book: electrochemical potential. For a biological cell, taking B inside and A outside makes the voltage term use the existing membrane potential convention.

Movement of proteins and lipids between cellular compartments by vesicles, tubules, membrane fusion and selective retrieval.

A conserved endosomal cargo-sorting complex participating in retrieval pathways. Associated sorting machinery varies between organisms and cargos.

Initial selective capture of a transport carrier before close membrane docking and fusion.

An ion-powered rotary machine driving bacterial flagellar motion. Its energy source and conformational switch are separable: inward ion flux can power either rotational direction.

A minimal bacterial flagellar motor model combines angular velocity $\omega >0$ with bound and detached probabilities $P_A+P_B=1$. With detachment rate $k_0$ in the first half-turn and attachment rate $k_0$ in the second, the periodic stationary bound probability is $P_A=\alpha e^{-\beta \theta }$ on $0\le \theta \le \pi$, where $\alpha =(1+e^{-\pi \beta }{)}^{-1}$, and $P_A(\theta +\pi )=1-P_A(\theta )$. The one-way switching windows describe externally driven kinetics rather than detailed balance with a passive heat bath.

For the driven two-state rotary motor with a triangular bound potential of height $U_0$, the torque scale is $T_0=U_0/\pi$. Differentiating the phase-averaged torque gives ${\overline{T}}^{\prime }(s)=T_0[2P_A(s)-1]/\pi$. The decreasing crossing of $P_A=1/2$ therefore gives $s_{\max }=\ln (2\alpha )/\beta$ and ${\overline{T}}_{\max }=T_0\ln \cosh (\pi \beta /2)/(\pi \beta )$. The slow-speed limit is $T_0/2$; the fast-speed limit vanishes as $T_0\pi k_0/(8\omega )$.

A switch-complex component participating in organization of the bacterial flagellar rotor's cytoplasmic ring.

A switch-complex protein involved in reception of phosphorylated CheY and control of rotational bias.

A rotor protein involved in stator contact and torque generation. Changes in its organization contribute to directional switching without reversal of ion flow.

A stator-associated protein contributing proton-coupling machinery and anchorage to the cell envelope in the familiar proton-driven motor.

A stator protein contributing to ion-coupled force generation in proton-driven bacterial flagellar motors.

An ATP-regulated anion channel whose phosphorylation-dependent activity contributes to epithelial chloride transport. Its activation downstream of cyclic AMP is important in cholera-associated secretion.

A substance facilitating proton transfer across a membrane and dissipating the proton electrochemical gradient. It can uncouple respiration from ATP synthesis without directly inhibiting every electron-transfer reaction.

A sodium-coupled glucose transporter important in apical intestinal uptake. The sodium gradient maintained by basolateral pumping powers uptake against a glucose gradient.

A bacterial sugar-uptake system coupling transport to phosphorylation using a phosphoenolpyruvate-derived phosphoryl relay. Its phosphorylation state also contributes to carbon-source regulation.

The glucose-specific enzyme-IIA component of the bacterial phosphotransferase system. Its dephosphorylated form can bind lactose permease, causing inducer exclusion.

The LacY transporter importing lactose in Escherichia coli. Binding of dephosphorylated EIIA glucose can inhibit transport and reduce inducer accumulation.

ESCRT components help sort membrane cargo and remodel membranes, including formation of internal vesicles in an endosome. Ubiquitin can be a cargo-selection signal, but recognition and the membrane-remodelling steps are distinct operations.

COPI is a coat system involved in retrieval toward the endoplasmic reticulum and traffic associated with the Golgi apparatus. Its coatomer assembly and ARF-family recruitment differ from COPII, while sharing membrane recruitment and cargo-selection principles.

COPII is a vesicle-coat system associated with export from the endoplasmic reticulum. Sar1 recruits an inner Sec23/Sec24 layer, and Sec13/Sec31 forms an outer coat. Cargo selection and membrane deformation are coupled but experimentally separable.

An adaptor protein complex couples membrane identity and cargo recognition to coat assembly. Coincident recognition of a membrane determinant, a cargo motif and coat partners helps restrict productive assembly to the appropriate site.

alternating access is a transport mechanism in which a substrate-binding region becomes accessible to opposite sides of a membrane in different states. A continuously open pore is avoided; coupling the state sequence to a driving process permits active transport.

Passive transport is net movement down the transported substance's electrochemical gradient without coupling that movement to another source of free energy. Facilitated diffusion is passive transport through an ion channel or a passive carrier protein. Providing such a route can speed passage, but does not turn an uphill transfer into a downhill one. For an uncharged solute under the ideal conditions above, net passive transport runs from higher to lower concentration; for an ion, the membrane potential also matters. See Cooper, The Cell: transport of small molecules.

In the figure, stipulate eight neutral solute tracers in A and two in B, with equal fixed volumes, identical surroundings, and no imposed electric potential difference. The concentration ratio is $c_{\mathrm{A}}/c_{\mathrm{B}}=8/2=4$, so A-to-B transfer has $\Delta \tilde{\mu }=-RT\ln 4\simeq -1.38629RT$, or $-3.458\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ at the assumed temperature $T=300\mathrm{K}$. If this solute can cross, thermodynamic equilibrium has equal expected concentrations: five tracers per side on average. Individual crossings occur in both directions; an observed ten-particle snapshot need not contain exactly five on each side. These stipulated counts illustrate a driving force, not a measured flow rate or a prediction of the time to equilibrate.

For a separate illustrative ion calculation, keep $c_{\mathrm{A}}/c_{\mathrm{B}}=4$, but assign $z=+1$ and $V_{\mathrm{A}}-V_{\mathrm{B}}=-60\mathrm{mV}$ at $300\mathrm{K}$. Using $R\simeq 8.31446\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$ and $F\simeq 96485.33\mathrm{C}\mathrm{mo}{\mathrm{l}}^{-1}$, B-to-A transfer gives $RT\ln 4-0.060F\simeq 3.458-5.789=-2.331\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$. Thus an appropriate open ion channel can permit passive transport toward the higher concentration: the electrical contribution outweighs the opposing chemical potential contribution. This stipulated membrane potential is not inferred from the tracer counts.

Facilitated diffusion is passive transport through a membrane transport protein, such as an ion channel or carrier protein, down the relevant electrochemical gradient. The protein supplies a route; it does not by itself provide free energy for uphill transfer.

In an alternating-access model a protein binding site faces either side of a membrane; both empty and substrate-bound carriers can switch orientation. Mass-action binding on each side and conformational transition rates give a four-state carrier cycle coupled to two free-substrate pools. Conservation counts all four carrier states as one protein each and counts bound substrate with the free-substrate pools. Equal source and sink fluxes preserve the total substrate while maintaining throughput.

With fixed interior substrate and finite carrier protein, symmetric alternating-access kinetics produce a monotone concave transport curve of the displayed form, with $A,C,D>0$. The flux reverses at equal free concentrations and approaches the finite limit $C/A$ for abundant exterior substrate. Its plateau reflects finite carrier abundance and finite turnover, rather than unlimited transport proportional to the concentration difference. The protein total multiplying the flux is distinct from the total substrate.

Osmosis is net movement of water across a boundary whose permeability differs for water and relevant solutes. The direction depends on the chemical potential of water, including both dissolved material and pressure. A cell wall can let water reach the cell membrane, yet oppose expansion through turgor pressure. Zero net movement does not mean individual water molecules have stopped crossing.

Imagine pure water on side A and a chemical solution on side B of a semipermeable membrane. At equal temperature and pressure, dissolved particles lower the chemical potential of water in B, and net water transfer initially favors A to B. Retaining the solutes lets water redistribute without simply mixing everything. Increasing B's pressure opposes entry; enough pressure gives thermodynamic equilibrium with unequal solute concentrations. This is why “water moves toward more solute” requires a pressure condition.^[ref]

At thermodynamic equilibrium, the chemical potentials of exchangeable water are equal. Individual water molecules still move and cross in both directions, with equal mean opposing transfer rates and zero net flux. A stationary liquid level alone does not prove this: a blocked route can also give no visible transfer. Selective permeability provides a route; water potential supplies the direction, and neither alone determines the speed. The cell wall, turgor pressure and plasmolysis articles show why a living boundary's mechanical support and its selective permeability must be distinguished.

Water potential expresses the chemical potential of water relative to a stated pure-water reference in units of pressure. For a nearly incompressible liquid at fixed temperature, use a constant reference molar volume ${\bar{V}}_w$ and writeThe reference is pure water at the same temperature and a chosen pressure $P_0$. Both sides must use the same reference. Neglecting height differences and interactions with solid surfaces, the approximation isHere $a_w$ is the dimensionless thermodynamic activity of water, $R$ the molar gas constant, and $T$ the absolute temperature. The natural logarithm is therefore applied to a dimensionless quantity. The pressure contribution ${\Psi }_p=P-P_0$ increases with pressure; the solute potential ${\Psi }_s$ is negative when $a_w<1$. Across an available passive water route, net water transfer favors higher to lower water potential. The familiar plant description adds effects of height and interactions with solid surfaces when they matter; they are omitted in the two liquid compartments here.^[ref]

Adding solute and raising pressure can thus cancel in water potential while each contribution remains nonzero. Equality of water potential does not imply equality of pressure or of every solute concentration. In a supported plant cell, the cell wall permits a positive turgor pressure to balance a negative solute potential; plasmolysis illustrates the opposite loss of support after water leaves.

Solute potential is the contribution to water potential from dissolved material, with pure water assigned zero. In the nearly incompressible description, ${\Psi }_s=(RT/{\bar{V}}_w)\ln a_w$. In a dilute ideal chemical solution, ${\Psi }_s\simeq -RT{c}_{\mathrm{osm}}$, where $c_{\mathrm{osm}}$ is the osmotic concentration of dissolved particles. Thus solute potential is negative while the corresponding osmotic pressure is positive: ${\Psi }_s\simeq -\Pi$. This sign difference distinguishes a contribution to water potential from the compensating mechanical pressure.^[ref]

Osmotic pressure is the excess pressure needed on a chemical solution to maintain osmotic thermodynamic equilibrium with pure solvent across a semipermeable membrane. For a nearly incompressible solvent, equality of chemical potential gives $\Pi \simeq -(RT/{\bar{V}}_w)\ln a_w$. It is an equilibrium property; an observed initial swelling rate is a different measurement.^[ref]

For a dilute ideal chemical solution of retained particles,Here $n$ is the amount of substance of independently dissolved solute particles, measured in moles; $V$ is the final volume of chemical solution, not the starting volume of pure solvent. Count the resulting ions separately if a solute dissociates; do not multiply twice if $n$ already counts them. With $c_{\mathrm{osm}}$ in $\mathrm{mol}{\mathrm{m}}^{-3}$ and $R$ in $\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$, $RT{c}_{\mathrm{osm}}$ has units $\mathrm{J}{\mathrm{m}}^{-3}=\mathrm{Pa}$. This resembles the ideal gas law, although the particles here are dissolved in a liquid.

The assumptions are uniform temperature, a dilute ideal chemical solution, a nearly incompressible solvent, and a semipermeable membrane that retains the counted solutes while passing solvent. Association, incomplete dissociation and interactions change the relation between added material and independent particles; nonideal thermodynamic activities invalidate simple particle counting. The formula does not describe a concentrated brine merely because the apparatus contains a membrane.

A short derivation explains why particles are counted. In an ideal mixture, $a_w=x_w=n_w/(n_w+n)$, with $n_w$ the amount of substance of water. If $n\ll n_w$, then $-\ln x_w=\ln (1+n/n_w)\simeq n/n_w$ and $V\simeq n_w{\bar{V}}_w$. Substitution gives $\Pi \simeq RTn/V$. This is a dilute-limit approximation, not a claim that water molecules stop moving or that dissolved particles are a gas.