Water, noncovalent interactions and biological conditions

The surrounding water belongs to the molecular question

A protein, membrane or nucleic-acid segment is studied in an environment: solvent, ions, temperature and chemical partners. Changing that environment can change the relative populations of chemical forms and assemblies. A molecular formula alone does not specify the conditions in which a biological function is observed.

Water has polar O–H bonds and can participate in hydrogen bonding. Its interactions with dissolved ions and polar groups help explain hydration. These interactions do not imply that every ionic solid or polar molecule dissolves completely. Solubility depends on the whole transfer between states, including the starting material and solution.

Noncovalent interactions include electrostatic, hydrogen-bonding and dispersion contributions. Their effects in solution must include competing interactions with the solvent. A contact drawn between two biomolecules is not by itself a measured binding free energy. Likewise, aggregation of nonpolar regions reorganises both solute and solvent; it should not be represented as a newly created covalent bond.

Count ions and squared charges separately

Concentration-based ionic strength is I_c=½Σc_i z_i², where c_i is the amount concentration of free ion i and z_i its signed charge number. I_c has concentration units. Squaring the charge makes a doubly charged ion contribute four times its concentration to the sum. Electroneutrality instead uses signed first powers, Σc_i z_i.

In an invented complete-dissociation inventory, 0.100 mol L⁻¹ NaCl supplies 0.100 mol L⁻¹ each of Na⁺ and Cl⁻. Thus I_c=½(0.100+0.100)=0.100 mol L⁻¹. The corresponding 0.100 mol L⁻¹ CaCl₂ inventory supplies 0.100 mol L⁻¹ Ca²⁺ and 0.200 mol L⁻¹ Cl⁻, giving I_c=½(4×0.100+0.200)=0.300 mol L⁻¹. Both inventories have zero total charge.

The calculation stipulates dissociation and excludes complexes. It is not evidence that activity coefficients equal one at these concentrations. Ionic strength, total independent ion concentration and salt formula-unit concentration are different quantities. A biological medium can contain several salts, ion-binding partners and other constituents.

A fixed pH sets a ratio under an assigned equilibrium model

Consider the hypothetical acid equilibrium HA ⇌ H⁺ + A⁻. Assign pK_a=6.80, externally maintained pH=7.00 and total acid-form concentration C_T=c_HA+c_A=20.0 mmol L⁻¹. For this separate ideal model use activity coefficients one and standard concentration c°=1 mol L⁻¹. pH is defined from hydrogen-ion activity; the concentration approximation is an explicit assumption.

The form ratio is r=c_A/c_HA=10^(pH−pK_a)=1.58489319. Normalizing the two forms gives f_HA=1/(1+r)=0.38686318 and f_A=r/(1+r)=0.61313682. Concentrations are therefore 7.7372636 and 12.2627364 mmol L⁻¹. Their sum preserves the assigned acid inventory.

This is a controlled-pH speciation calculation, not the pH of an unbuffered acid solution. The maintained environment supplies the required proton/charge accounting; its buffer and counterions are not quantified here. The plot varies controlled pH with the same assigned pK_a and model. It does not describe a real cell, universal biological pH or a measured titration.

Activity and concentration are connected through a model

For a specified solvent standard, chemical potential can be written μ=μ°+RT ln a, where a is dimensionless activity, T absolute temperature and R the molar gas constant. Comparing two states under that same standard gives Δμ=RT ln(a₂/a₁). Neither a quantity with units nor an unexplained concentration number belongs directly inside this logarithm.

An original example stipulates solvent activity a₂=0.980, reference a₁=1 and T=298.15 K. Using R=8.31446261815324 J mol⁻¹ K⁻¹ gives Δμ=−50.0816433 J mol⁻¹. The negative sign means the assigned state has lower solvent chemical potential than that reference.

The activity was supplied rather than estimated from concentration. This difference is not the free energy of a particular hydrogen bond, a protein fold or a complete cellular process. The standard state and thermodynamic model make those different questions distinguishable.

Osmotic counting uses independent solute entities

For an ideal dilute solution relative to pure solvent at the same temperature on the other side of a suitable boundary permeable to solvent but excluding the stated solutes, the osmotic-pressure model is π=c_particle RT. Here c_particle counts independent dissolved entities in mol m⁻³. A polymer chain is one such entity in the stipulated dilute nonassociating model; its repeat units do not each become independent particles merely because they can be counted.

Assign total particle concentrations 10.0 and 20.0 mmol L⁻¹ at 300 K. Convert them to 10.0 and 20.0 mol m⁻³. The model gives 24943.3879 and 49886.7757 Pa, approximately 24.9 and 49.9 kPa. Doubling the specified independent-particle concentration doubles this ideal pressure.

These particle totals are given independently of the earlier salt examples. Dissociation, association, interactions and membrane permeability can change the relevant accounting. The ideal equation alone does not predict a biological cell volume or the behavior of a selectively permeable membrane with several transported solutes.

Report the conditions before attributing a function

Hydration, protonation and ionic conditions can change how molecular groups interact. A comparison of biological molecules therefore needs the actual forms present and the conditions of the measurement. A bare structural drawing and a salt label leave several quantities unresolved.

Keep ionic strength, acid-form concentration, solvent activity and osmotic particle concentration in separate columns of the reasoning. Each has its own model and units. Agreement with one of these hypothetical calculations would not validate the other three or establish a biological function.

Worked examples

Try it yourself

1. A fully dissociated, noncomplexing mixture contains 0.0500 mol L⁻¹ NaCl and 0.0200 mol L⁻¹ CaCl₂. List free ions and find I_c.

   Show the answer and reasoning

   Na⁺ 0.0500, Ca²⁺ 0.0200 and Cl⁻ 0.0900 mol L⁻¹; I_c=0.110 mol L⁻¹.

   Chloride is 0.0500+2×0.0200. Half the squared-charge sum is ½(0.0500+4×0.0200+0.0900).

2. With pK_a=6.80, controlled pH=6.50 and C_T=30.0 mmol L⁻¹, calculate both form concentrations.

   Show the answer and reasoning

   HA 19.9841827 and A⁻ 10.0158173 mmol L⁻¹.

   r=10^−0.30=0.50118723; fractions 0.66613942 and 0.33386058 preserve the new total. This does not calculate unbuffered pH.

3. At 310 K with the same solvent standard, calculate μ(a=0.950)−μ(a=1).

   Show the answer and reasoning

   −132.207615 J mol⁻¹.

   RT ln(0.950) uses a dimensionless activity ratio. The assigned activity is lower than the reference.

4. An ideal dilute solution has total independent particle concentration 15.0 mmol L⁻¹ at 310 K. Find π in Pa and kPa.

   Show the answer and reasoning

   38662.2512 Pa, or 38.6622512 kPa.

   Convert to 15.0 mol m⁻³ before applying π=c_particle RT. Keep formula units or polymer residues separate from independent entities.

What to carry forward

• Water and dissolved constituents are part of the chemical environment.
• Ionic strength uses squared ion charges; electroneutrality uses signed charges.
• Fixed-pH fractions require a declared equilibrium convention.
• Activity in a logarithm is dimensionless.
• Ideal osmotic counting distinguishes independent entities from constituent units.