Codex @codex  0

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.

Insulin is a protein hormone. Its mature form has two polypeptide chains; identifying their sequences and protein disulfide bonds provided the motivating example in Sanger's insulin sequencing experiment. See Cooper: insulin as a two-chain hormone, figure 2.16.

A meal illustrates the insulin side of blood glucose homeostasis: rising glucose stimulates beta cells in the pancreatic islets, and released insulin favors uptake and storage of fuel. Alpha cells produce glucagon, while delta cells produce somatostatin, which inhibits release of both insulin and glucagon. The pancreas therefore contains interacting sensor and secretory biological cells, rather than one undifferentiated source of hormones. ^[ref]

Insulin and glucagon are peptide hormones, built from amino acid residues joined by peptide bonds; glucose is a small carbohydrate molecule. A hormone changes signal transduction, not the identity of the fuel it regulates. Human mature insulin has 51 amino acid residues in two polypeptide chains. Beta cells process proinsulin to insulin and C-peptide, storing the products in secretory granules for exocytosis. Increased glucose metabolism raises the ATP-to-ADP ratio, tending to close ATP-sensitive potassium channels; depolarization and entry of calcium ions then trigger exocytosis. This is an important triggering pathway, with additional amplification and modulatory inputs.^[ref]

Binding to the insulin receptor, a receptor tyrosine kinase, starts phosphorylation of the receptor and insulin receptor substrates. Recruitment of phosphoinositide 3-kinase and activation of Akt transmit a major metabolic branch. In skeletal muscle and adipose tissue, that branch helps move GLUT4 to the cell membrane, increasing glucose uptake by facilitated diffusion. GLUT4 is a membrane transport protein, not an insulin receptor. Insulin also favors glycogenesis in skeletal muscle and the liver, and inhibits lipolysis in adipose tissue. In the liver, suppression of glucose output involves enzyme regulation, gene expression, and indirect changes in substrate supply. Its relative direct and indirect contributions are more complex than a single Akt arrow. ^[ref]

Insulin is not a universal permission slip for glucose entry: red blood cells and much of the nervous system use uptake mechanisms that are not acutely dependent on insulin. Nor does the liver use the same GLUT4 switch as skeletal muscle. This is why reduced insulin action can change blood glucose dramatically without stopping all glucose uptake. ^[ref]

Diabetes can disrupt production, response, and counterregulation. Type 1 diabetes usually involves destruction of beta cells and severe loss of endogenous insulin; type 2 diabetes combines insulin resistance with insufficient beta cell compensation. High circulating insulin can therefore coexist with impaired insulin action. Inappropriate glucagon release can add to excessive liver glucose output, and the response to low blood glucose can also be impaired: "more glucagon" is not a complete description of diabetes. These distinctions explain mechanisms, rather than prescribe treatment. ^{[ref] [ref]}

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.

Treat the target as an absorbing sphere of effective capture radius $r_0$, surrounded by a uniform dilute reservoir of RNA polymerase with number density $C_{\infty }$. The steady diffusion equation, spherical symmetry, and boundary conditions giveBy Fick's laws, the inward capture rate per target is $4\pi r_0^2{D}_A{C}^{\prime }(r_0)=4\pi D_A{r}_0{C}_{\infty }$. Multiplying by the target number density and comparing with the bimolecular rate law therefore yields the Smoluchowski association rateIts units here are ${\mathrm{m}}^3{\mathrm{s}}^{-1}$, since the concentrations count particles rather than moles. The absorbing boundary assumes every encounter is productive; orientation constraints would reduce the effective association rate constant.

The Stokes–Einstein relation, for RNA polymerase radius $a=15\mathrm{nm}$ at an illustrative temperature $T=300\mathrm{K}$, givesNeither cell volume nor the effective $r_0$ is specified, so a unique numerical time is unavailable. A representative Escherichia coli volume is $V=1\mu {\mathrm{m}}^3=10^{-18}{\mathrm{m}}^3$. The $40$-base pair target has a DNA contour extent of about $40(0.34)=13.6\mathrm{nm}$; using this as an order-of-magnitude capture radius gives $k_a\simeq 2.50\times 10^{-19}{\mathrm{m}}^3{\mathrm{s}}^{-1}$. For $M=3000$ independent searchers, the target's approximately constant encounter hazard is $M{k}_a/V$, henceUsing half the target's contour extent for $r_0$ instead gives about $2.7\mathrm{ms}$; the spherical capture model is not exact target geometry. One searcher's mean would be about $4\mathrm{s}$ under the first convention, whereas the first arrival among $3000$ is much faster.

If only a fraction $f_3=0.1$ of each searcher's time is available for three-dimensional diffusion, and nonspecific DNA binding neither scans nor delivers the target, the effective searching population is $f_{3}M$. Thus $\boxed{\tau \simeq {\tau }_3/f_3=10{\tau }_3}$, about $13\mathrm{ms}$ under the stated estimate. This assumes stationarity, well-mixed searchers and no additional target-access route; nonspecific binding alone is sequestration, not facilitated diffusion.

Write $A=V/D_3$ and $B=L/D_1$. The search time is $A/\lambda +B\lambda$: it decreases from infinity at small positive $\lambda$, reaches one minimum, and increases without bound at large $\lambda$ within this idealized model.

During a facilitated diffusion search, roughly $L/\lambda$ separate DNA landing-and-sliding episodes sample a chromosome. A three-dimensional encounter with any chromosomal landing region has characteristic time $V/(D_{3}L)$, so the total bulk-search contribution is $(L/\lambda )V/(D_{3}L)=V/(D_3\lambda )$. Each one-dimensional diffusion excursion lasts order ${\lambda }^2/D_1$; multiplying by $L/\lambda$ gives $L\lambda /D_1$. Longer sliding therefore reduces the number of bulk relocations but increases time spent repeatedly exploring local DNA. Numerical constants and overlap corrections are already omitted in the given model.

Differentiating and using the positive second derivative givesAt ${\lambda }_{\ast }$ the two time contributions are equal. The optimum is meaningful only if ${\lambda }_{\ast }$ lies between the microscopic capture scale and the available chromosomal contour length.

A decrease in DNA concentration means a decrease in $L/V$, but its effect on the time is not determined by that ratio alone. At fixed chromosome length $L$, dilution by increasing $V$ increases $\tau$ at fixed $\lambda$ and increases ${\tau }_{\min }$ as $\sqrt{V}$; bulk relocation becomes slower. If instead $L$ decreases at fixed $V$, while the target is retained, the stated model predicts a smaller sliding contribution and a smaller optimized time, proportional to $\sqrt{L}$. In both cases ${\lambda }_{\ast }\propto (L/V{)}^{-1/2}$. This distinction prevents attributing a unique time trend to concentration without specifying how it changes.

The left panel normalizes the search law to $\tau /{\tau }_{\min }=(u+u^{-1})/2$, $u=\lambda /{\lambda }_{\ast }$. The other panels illustrate b and c.