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Deoxyribonucleic acid, or DNA, is a nucleotide polymer that stores biological information in its nucleotide sequence. The crucial variable is order: two DNA molecules can contain the same numbers of each nucleobase yet carry different nucleotide sequences. This makes a physical molecule an inheritable record, rather than merely a mixture of ingredients. Genes are functional regions of that record; the whole DNA nucleotide sequence is not one long gene. See NHGRI: DNA fact sheet.

Each DNA strand has a backbone of alternating deoxyribose sugars and phosphate groups joined through covalent bonds. Attached nucleobases supply the familiar four-letter alphabet: adenine A, thymine T, guanine G, and cytosine C. In ordinary double-stranded DNA, base pairing matches A with T and G with C. The backbone connects successive nucleotides along a strand; base pairing connects partners across strands. Keeping these two connections distinct explains how strands can separate while each retains its own nucleotide sequence.

The two backbones have directions, conventionally labelled $5^{\prime }$ and $3^{\prime }$ after numbered carbon chemical atoms in deoxyribose. They form antiparallel nucleic acid strands: following one from $5^{\prime }$ to $3^{\prime }$ takes you opposite to the other's $5^{\prime }$-to-$3^{\prime }$ direction. For the chosen DNA nucleotide sequence $5^{\prime }\text{-}\mathrm{ATGCC}\text{-}{3}^{\prime }$, write its partner directly underneath as $3^{\prime }\text{-}\mathrm{TACGG}\text{-}{5}^{\prime }$. Now read that partner from its own $5^{\prime }$ end: its reverse complement is $5^{\prime }\text{-}\mathrm{GGCAT}\text{-}{3}^{\prime }$. Complementing letters and reversing their order are separate operations. See Alberts et al., Molecular Biology of the Cell: the structure and function of DNA.

The diagram's rails represent sugar–phosphate backbones and its coloured blocks represent nucleobases, not individual chemical atoms. Reading left to right gives five matched positions: A–T, T–A, G–C, C–G, C–G. There are two A–T-type base pairs and three G–C-type base pairs; the complete two-strand example contains ten nucleotides. Reversing the bottom row changes the reading convention, not the physical partners. Real double-stranded DNA winds into a nucleic acid double helix; the flattened ladder emphasizes base pairing and direction without pretending to show its three-dimensional geometry.

The motivating consequence is copying by a template. After strands separate, each existing DNA strand specifies which incoming nucleotides should accompany it through base pairing. DNA replication uses enzymes to build complementary strands, allowing the record to persist through successive biological cells. Base pairing supplies a selection rule, while enzymes perform the chemistry: a matching diagram alone is not a complete copying mechanism. The connection between structure and copying was already central to the model published by Watson and Crick on 25 April 1953. See Watson and Crick, Molecular Structure of Nucleic Acids, Nature 171, 737–738 (1953).

Stored information needs interpretation. Some DNA regions specify proteins through RNA; others contribute to functional RNA or regulate gene expression, and some have no known function. Non-coding DNA means sequence that does not specify a protein's sequence of amino acids, not automatically useless sequence. Nor does a DNA nucleotide sequence alone determine an organism's fate: phenotypes also depend on biological context and environment. See NHGRI: non-coding DNA. Its double-stranded form is often modeled mechanically as a worm-like chain, with elasticity and effective excluded volume depending on solvent conditions.

For a single-site solvent and polymer length $N$, expand $(1-\varphi )\ln (1-\varphi )=-\varphi +{\varphi }^2/2+{\varphi }^3/6+O({\varphi }^4)$ in Flory–Huggins solution theory. The displayed result follows. Its linear term changes reference chemical potentials; its quadratic term sets the effective two-segment excluded volume. The latter vanishes at the theta temperature, while the positive cubic term remains. Taking $\varphi f^{\prime }-f$ yields the osmotic pressure in Flory-Huggins solution theory, whose cubic coefficient is $1/3$ rather than the free energy's $1/6$.

For a free three-dimensional worm-like chain with contour length $L$ and persistence length ${\ell }_p$, the unit-tangent correlation is $⟨\mathbf{t}(s)\cdot \mathbf{t}(s^{\prime })⟩=e^{-|s-s^{\prime }|/{\ell }_p}$. Since $\mathbf{R}={\int }_0^L\mathbf{t}(s)ds$, integrating this correlation givesFor $L\gg {\ell }_p$, the leading term is $2{\ell }_{p}L$, the ideal chain law with Kuhn length $2{\ell }_p$. For $L\ll {\ell }_p$, a Taylor series gives $⟨R^2⟩=L^2-L^3/(3{\ell }_p)+O(L^4/{\ell }_p^2)$, approaching a rigid rod. This ideal semiflexible-chain model omits long-range excluded volume; sufficiently long dilute chains in a good solvent can cross over to swollen-chain statistics.

The Flory approximation balances the entropic cost of expanding an ideal chain against a mean-field excluded volume interaction. For $N$ segments of length $b$ in $d$ spatial dimensions, minimizing $R^2/(N{b}^2)+v{N}^2/R^d$ gives $R^{d+2}\sim v{N}^3{b}^2$. This yields the scaling exponents $3/5$ in three dimensions and $3/4$ in two dimensions. It is a scaling approximation, not an exact numerical free energy or proof of critical exponents; confinement replaces $R^d$ by the accessible coil volume.

The Flory interaction parameter measures the excess free-energy cost of unlike contacts in Flory–Huggins solution theory. For a simple lattice of coordination number $z_c$ and contact energies ${ϵ}_{ps}$, ${ϵ}_{pp}$ and ${ϵ}_{ss}$,Real effective parameters may also contain entropic contributions and composition dependence. With site volume $v_0$ and composition-independent $\chi$, the effective two-segment excluded volume is $v_{\mathrm{eff}}=v_0(1-2\chi )$. Positive, zero and negative $v_{\mathrm{eff}}$ correspond respectively to a good solvent, the theta temperature condition and a poor solvent. In particular, $\chi =1/2$ means cancellation of leading repulsive and attractive contributions; it does not mean that every molecular interaction vanishes.

A good solvent favors contact with polymer segments strongly enough that the effective excluded volume is positive. A long dilute flexible chain swells relative to an ideal chain; the three-dimensional size exponent is approximately $0.588$, often replaced by $3/5$ in a scaling estimate. This designation depends on the polymer, solvent and temperature, rather than on the solvent alone.

A hard-sphere gas is a classical model of identical impenetrable spheres, each of diameter $\sigma$ and radius $\sigma /2$. Its pair potential is $u(r)=+\infty$ for centre separation $r<\sigma$ and $u(r)=0$ for $r\ge \sigma$; the convention at exact contact has no effect on the integral. It isolates the effect of particle size without an attractive potential energy.

The upper-left panel is a cross-section of a three-dimensional forbidden region: the second centre cannot enter a sphere of radius $\sigma$ around the first centre. The particles themselves have radius $\sigma /2$. Consequently, with particle volume $v_{\mathrm{s}}$ and pair excluded volume $v_{\mathrm{ex}}$,The last equality follows directly from the Mayer f-function: $f=-1$ throughout the forbidden volume and zero outside it. Confusing particle radius with contact separation changes the answer by a factor of eight; omitting the unordered-pair factor changes it by a further factor of two.

Define the packing fraction $\eta =n{v}_{\mathrm{s}}$, where $n$ is number density. Then the virial expansion beginsFor example, at the toy value $\eta =0.01$, the leading correction predicts $Z\approx 1.04$; the omitted terms still matter if accuracy beyond this approximation is needed. The classical second virial coefficient is independent of temperature because the geometry of forbidden configurations does not change when the gas is warmed. The pressure itself still scales with temperature at fixed number density.

This model makes the positive correction particularly clear: eliminating overlapping configurations changes the dependence of the available configuration space on volume, even though the allowed configurations all have zero interaction energy. The figure's other panels add a toy attractive shell and show how its Boltzmann factor competes with that geometry. The hard-sphere gas is a useful reference model, while a low-order virial expansion remains a dilute-gas approximation rather than a dense sphere packing formula.

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The Van der Waals equation connects a dilute gas model to the possibility of a liquid-gas critical point. Its usefulness includes showing how attractions and excluded volume can change an equation of state; its simple constants do not make it an exact model for every fluid.

Choose a hypothetical fluid with molar constants $a_m=0.360\mathrm{Pa}{\mathrm{m}}^6\mathrm{mo}{\mathrm{l}}^{-2}$ and $b_m=4.00\times 10^{-5}{\mathrm{m}}^3\mathrm{mo}{\mathrm{l}}^{-1}$, using $R=8.314462618\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$. These stipulated constants are not a fit to carbon dioxide. The molar critical formulas giveAt $T=360\mathrm{K}$ and $v_m=240\mathrm{c}{\mathrm{m}}^3\mathrm{mo}{\mathrm{l}}^{-1}$, first convert $v_m$ to $2.40\times 10^{-4}{\mathrm{m}}^3\mathrm{mo}{\mathrm{l}}^{-1}$. The repulsive-volume term and attractive correction areso $p=8.716033\mathrm{MPa}$. The ideal gas law would instead give $RT/v_m=12.471694\mathrm{MPa}$. The excluded volume raises pressure relative to that ideal value, but attraction more than offsets the increase here. Neither correction alone describes this example.

For $n_{\mathrm{mol}}=0.500\mathrm{mol}$, the same state occupies $V=n_{\mathrm{mol}}{v}_m=120\mathrm{c}{\mathrm{m}}^3$. The total-volume formula gives $n_{\mathrm{mol}}{b}_m=20\mathrm{c}{\mathrm{m}}^3$ and the same $8.716033\mathrm{MPa}$. Halving the sample has halved both its physical volume and its excluded volume, while retaining its pressure. Conversely, if the two critical measurements $T_c,p_c$ are used to determine model constants, inversion givesAn independent critical molar volume then tests the prediction $v_{m,c}=3R{T}_c/(8p_c)$; agreement with two fitted numbers does not ensure agreement with the third.

At $t=0.9$, the Van der Waals liquid-gas coexistence example below becomes $T\simeq 288.6537\mathrm{K}$, $p_{\ast }\simeq 5.391653\mathrm{MPa}$, $v_{m,l}\simeq 72.40823\mathrm{c}{\mathrm{m}}^3\mathrm{mo}{\mathrm{l}}^{-1}$ and $v_{m,g}\simeq 281.8611\mathrm{c}{\mathrm{m}}^3\mathrm{mo}{\mathrm{l}}^{-1}$. For a mean molar volume of $120\mathrm{c}{\mathrm{m}}^3\mathrm{mo}{\mathrm{l}}^{-1}$, the lever rule gives gas mole fraction $f_g\simeq 0.227219$. Its volume fraction is instead $f_g{v}_{m,g}/v_m\simeq 0.533703$: less than one quarter of the molecules occupy more than half the volume. The flat equilibrium pressure corresponds to changing these proportions. It does not describe a homogeneous sample sliding along the unstable loop.

For slender hard rods of length $L$ and diameter $D\ll L$, the leading pair excluded volume is the displayed function of their mutual angle $\gamma$. Alignment lowers mutual obstruction and increases positional entropy, while narrowing the orientation distribution lowers orientational entropy. Their competition can drive an isotropic-to-nematic transition without attractive forces. The natural concentration scale is $c{L}^{2}D$ of order one. Finite rod diameter, flexibility and interactions beyond the hard-core second-virial approximation change quantitative predictions.

Let $f(\varphi )$ be the mixing free energy per lattice site in Flory–Huggins solution theory, $v_0$ the site volume, and $\varphi$ the polymer volume fraction. The solvent mixing chemical potential is $f-\varphi f^{\prime }$, so its balance against the pure-solvent pressure contribution gives $\Pi =(\varphi f^{\prime }-f)/v_0$. For composition-independent $\chi$,The first term is the ideal chain translational contribution; the second encodes the effective two-segment excluded volume. At the theta temperature condition it vanishes but the positive third-order term remains. Treating $f$ as a free-energy density without dividing by $v_0$ loses the physical dimensions of osmotic pressure.

With wall effects neglected, $a_1=a$. A second hard disc cannot have its centre within distance $2r$ of the first centre. Its excluded volume in this two-dimensional setting is an excluded area $\pi (2r{)}^2=4a_d$, givingThis is exclusion of the second centre, rather than simply subtraction of the first disc's own area.

Let $b\simeq 2l_p$ be the Kuhn length, $N=L/b$ the number of statistical segments, and $v$ their effective excluded volume. We first assume a good-solvent, strongly confined self-avoiding walk, rather than an ideal chain. A confinement blob contains $g$ links and has diameter $D\sim b{g}^{3/5}$ in the simple $v\sim b^3$ scaling model. Therefore $g\sim (D/b{)}^{5/3}$. Distant blobs cannot freely pass through one another in a narrow tube, so the $N/g$ blobs form an axial string. The de Gennes confinement scaling is consequentlyThe free energy is of order $k_{B}T$ per blob, not the energy of stretching a straight elastic rod.

The role of solvent and chain thickness can be made explicit by a Flory approximation: an axial extension $X$ costs an entropic term $k_{B}T{X}^2/(N{b}^2)$, while the confined number density gives an interaction term $k_{B}Tv{N}^2/(X{D}^2)$. Minimizing their sum gives $X^3\sim v{N}^3{b}^2/D^2$, henceFor a thin statistical segment of effective diameter $d$, $v\sim b^{2}d$ instead gives $X\sim L(bd/D^2{)}^{1/3}$. Thus the simpler answer implicitly fixes the excluded volume; the DNA thickness and ionic environment can matter quantitatively.

Both blob arguments require $b\ll D\ll R_{\mathrm{free}}$ and $N/g\gg 1$, where $R_{\mathrm{free}}\sim b{N}^{3/5}$ in the simple good-solvent model. The printed comparison with $L$ alone does not ensure confinement relative to the unconfined coil. If $D\gg R_{\mathrm{free}}$, the tube barely perturbs the coil. An ideal chain without excluded volume has axial spread of order $\sqrt{Lb}$ and does not acquire the extensive extension of a nonoverlapping blob string. These are distinct physical regimes, not changes to a numerical prefactor.

A slit of height $D$ leaves two unrestricted directions. Its three-dimensional blobs now form a planar self-avoiding walk, whose size exponent is $3/4$. The polymer coil in slit confinement therefore has in-plane sizeThere is no uniquely preferred axial direction, and the chain is a coil rather than the one-dimensional blob string of a tube. Including a segment excluded volume $v$, the Flory approximation is $\mathcal{F}/(k_{B}T)\sim R_{\parallel }^2/(N{b}^2)+v{N}^2/(R_{\parallel }^{2}D)$. Its minimum satisfies $R_{\parallel }^4\sim v{N}^3{b}^2/D$, agreeing with the displayed scaling when $v\sim b^3$.

If “two-dimensional channel” instead means an intrinsically planar chain confined to a strip of width $D$, a blob is already two-dimensional: $D\sim b{g}^{3/4}$, and the blobs form a one-dimensional string. That different geometry gives $X\sim L(b/D{)}^{1/3}$. Stating which transverse direction is removed is essential. In either geometry the blob description requires many blobs and a width below the free-coil size.

The numerical contour length is $L=N{b}_0=10^{10}\times 10^{-8}\mathrm{m}=100\mathrm{m}$, and $k_{B}T=4.14\times 10^{-21}\mathrm{J}$. Begin with a three-dimensional freely jointed chain in which each chemical monomer is also an independent statistical link of length $b_0$. This neglects excluded volume, hydrodynamic effects on equilibrium statistics, and stretching of the chemical backbone.

An applied force $F$ gives a one-link canonical partition functionThe mean extension is $z=k_{B}T{\partial }_F\log Z_1^N=N{b}_0(\coth \xi -1/\xi )$. Expanding the Langevin function at small force gives $z=N{b}_0^{2}F/(3k_{B}T)+O(F^3)$. The three-dimensional entropic chain stiffness is thusThe unforced root-mean-square end-to-end distance is $\sqrt{N}{b}_0=10^{-3}\mathrm{m}$, not the contour length. This response is entropic straightening of a fluctuating coil. Axial stretching of a straight material rod instead has stiffness $EA/L$, and is a different phenomenon. Aqueous solvent does not by itself prove ideal-chain statistics: a good-solvent self-avoiding walk has a different coil size and susceptibility. The estimate above states explicitly the model under which the supplied link data suffice.

The extension must be compared with the actual contour length, $L=100\mathrm{m}$. For the illustrative $Q=0.2$ force and the persistence-length-corrected chain, $z/L\simeq 2.15\times 10^{-3}$ and $F{b}_K/(k_{B}T)\simeq 6.44\times 10^{-3}$. Both are small, so the small-force Langevin function expansion and entropic-extension approximation are consistent. Although the mean extension greatly exceeds the unforced millimetre-scale coil size, it remains far below the maximum contour length; that does not violate the model.

The material axial elongation at this force would be only $F/(EA/L)\simeq 5.0\times 10^{-5}\mathrm{m}$, illustrating that most of the predicted coil extension is orientation and straightening rather than stretching the backbone. Thus the order-of-magnitude entropic response is internally reasonable within the ideal-chain assumptions, but its numerical optical-force prediction is not unique. Solvent-dependent excluded volume, the unusually large stated contour length, actual trap geometry, and surface interactions all matter to an experimental interpretation.

The Clausius gas equation can be writtenComparison with the virial expansion givesFor a hard-sphere gas of atomic radius $r_0$, the pair potential is infinite when the center separation is below $2r_0$ and zero otherwise. ThusThis is four times the volume of one atom and is the per-particle excluded volume appearing in the dilute equation of state.

This is the second printed “(vi)”, concerning solvent temperature. The coil is expected to expand, so its radius of gyration increases. For the usual upper-critical-solution-temperature behavior of polystyrene in cyclohexane, heating above the stated theta condition improves solvent quality. Effective repulsion between segments favors more expanded conformations than the ideal coil.

In a Flory–Huggins solution theory description, the theta condition corresponds approximately to interaction parameter $\chi =1/2$, whereas a good solvent has a positive effective excluded volume. The asymptotic size dependence crosses toward $R_g\propto N^{\nu }$ with $\nu$ near $0.59$ for a flexible chain in three dimensions. These scaling ideas establish the expected direction, not a numerical expansion factor for a ten-degree temperature change. Its value would require measured solvent-interaction data or a calibrated crossover model.

For this self-assembly into a colloidal crystal, four useful elements are building blocks, interactions, environment and reversible dynamics. The building blocks here are nearly monodisperse polymer spheres; their shape and size distribution constrain the accessible packing. Their interactions include excluded volume, Van der Waals forces and electrostatic or steric stabilization, which set the free energy landscape. The aqueous environment supplies temperature, viscosity, ionic strength, confinement and concentration controls. Finally, Brownian motion and rotation must let particles encounter one another and rearrange or detach from incorrect contacts. Attractions that bind every collision irreversibly favour an amorphous aggregate rather than an annealed colloidal crystal. These four categories include both thermodynamic selection and the transport/kinetic access needed to realize it.

At the theta condition, effective excluded volume approximately cancel and a freely jointed ideal chain has $⟨R^2⟩=N{b}^2$. With the stipulated segment length $b=2a$,In a good solvent repulsive excluded volume swells the coil, giving $R\sim b{N}^{\nu }$ with $\nu \simeq 0.588$ in three dimensions, or the Flory approximation $\nu =3/5$. In a poor solvent effective segment attraction favours a compact globule, whose linear size scales as $b{N}^{1/3}$ at roughly fixed internal density. The latter are scaling laws with model-dependent prefactors. A real chemical monomer need not be a statistical Kuhn length; using $2a$ as the independent segment length is part of this question's idealized model.