Physical Chemistry of Drugs. Summary

This document contains statements, formulae, units and data tables which you are supposed to know.

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Objectives

Understand the underlying quantitative principles governing the behavior of therapeutic agents including:

stability and storage of a therapeutic formulation,
dissolution of drug crystals and powders and crystallization of drug metabolites
chemical and physical determinants of bioavailability including drug permeation to the blood stream and crossing of the blood brain barrier;
protonation and deprotonation, acid-base equilibrium and its role in permeability and solubility
tautomerization and sterioisomerization and their role in adverse effects
biological membranes, osmosis, and tonicity of drug solutions
drug-receptor binding thermodynamics, relationship between a therapeutic dose or concentration and the binding energetics
the molecular energetics and three dimensional structure of drug-protein complexes
off-target protein-drug binding including carriers and polypharmacology
kinetics including drug-binding kinetics, and drug diffusion

Understand research literature about therapeutics that uses the following methods: calorimetry, X-ray crystallography, NMR, chromatography, mass spectrometry.

Drugs

Drugs by size of the molecule of active ingredient
- Small molecules : from one atom (Xe) to about a four hundred non-hydrogen (a.k.a. heavy) atoms (the median is 24 heavy atoms, 99% of small drugs are between 5 to 70 heavy atoms)
- Peptide and natural products : peptides are aminoacid chains from 2 aminoacids to around 30 amino acids.
- Proteins : from 50 to thousands aminoacids (e.g. insulin, antibodies, growth factors )
- Vaccines
- Engineered viruses and reprogrammed/engineered cells (emerging)
By numbers from a set of active ingredients of the US-approved therapeutics as present in Drugbank on Jan 1, 2017:
- Small molecules: 1585
- 158 Unique Proteins/peptides (peptides and proteins from 8 amino acids to 2332 amino acids for Antihemophilic Factor).
- Median: 174 amino acids with 1400 atoms (compare with the median 24 atoms for small molecules)
- An average amino acid contributes about 8 heavy atoms.
Figure. Distribution of small molecule drugs by the number of non-hydrogen (a.k.a. heavy) atoms Source: approved drugs from Drugbank, www.drugbank.ca

Drugs by state of matter or phase
- gas (inhalers)
- liquid solutions
- micro-crystals or powder formulated with excipients

Units, scales, force, energy, temperature

The basic SI units : kilogram, meter, second
Temperature: measure of energy: Absolute temperature in Kelvin, K, and Celsius, C: K = C + 273.15
The kinetic energy of a point mass m at velocity v is ( mv² )/2
Energy in the SI system is measured in Joules ( J ≡ kg m² s^-2 ), see the previous statement
1 calorie = 4.184J is the energy needed to increase 1g of water by 1 degree, 1 kcal is the same for 1kg (1L) of water
1 Calorie (or food calorie, note the capital 'C') is 1kcal
1 electron-volt = 1.6×10⁻¹⁹ joule : 1eV of 1 mole of particles ≅ 100 kJ
Gas constant converts temperature to energy: R = 8.314 J K^-1 mol^-1 ; R = 1.986 cal K^{−1 mol⁻¹*} RT at 300K is 2.5 kJ/mol or 0.6 kcal/mol
Force: F = ma .Units of force: Newton: N ≡ kg m s^-2
Work ≡ Force * Distance . Newton*meter = kg m² s^-2 has the same units as energy.
Work changes the energy, without work the energy is conserved
Heat is another form of energy
The Avogadro number of a molecule is called one mole
N_A ≈ 6.* 10²³ (btw: 6=2*3 , exact value is 6.022 * 10²³ )

Moving Molecules, Gas Law

PV = nRT ; R = 8.314 J K^-1 mol^-1
Pressure is Force per unit area (P=F/A). SI Unit is pascal: 1Pa = Nm^-2 or ; 1bar = 10⁵ Pa ≅ 1atm = 760 mmMercury
P Δ V = P • Area Δ X = F Δ X = WORK .
Newton's law: F = m * a ( F and a are vectors)
Conservation of Momentum = P ≡ ∑ m v = const without force.
The total Momentum is conserved without external force, if the force is applied it is changed according to: ∂ P / ∂t = F (another form of Newton's 2nd law (3)).
Mean energy of molecules and velocities: 1/2 mv² = 3/2 k_B T for 1 molecule, use molar mass and R instead of k_B for 1 mole. The mean velocity is v_rms = Sqrt( 3RT / M ), M is molar mass.
Nitrogen (N₂ molecules) move at speeds around 500 m/s at room temperature or ≅ 1120 Miles/hour . 100 times heavier molecule moves 10 times slower.
1 m/s = 2.237 Miles/hour
Boltzman constant k_B is defined as the Gas constant divided by the Avogadro number: R/N_A

Thermodynamics

A very large number of moving molecules can be described with average variables (parameters), such as P,V,T,concentrations, energies, etc.
First law of thermodynamics: The increase in internal energy of a closed system is equal to the difference of the heat supplied to the system and the work done by it: ΔU = Q - W .
Second law of thermodynamics: Heat cannot spontaneously flow from a colder location to a hotter location.

Heat Capacity

Additional reading: a good review of heat capacity and degrees of freedom here

Heat capacity C ≡ Q/ΔT
C depends on the amount of substance and the process (e.g. constant volume or pressure)
C_V = ΔU/ΔT , C_V is the slope of the U(T) function
Another form of the equation: U_T = U_T₀ + C_V (T - T₀ ), with small ΔT or relatively constant C_P in the range of temperatures involved.
C_P = ΔH/ΔT , where H is enthalpy H ≡ U + PV , C_P is the slope of the H(T) function
Another form of the equation: H_T = H_T₀ + C_P (T - T₀ ) , with small ΔT or relatively constant C_P in the range of temperatures involved.
For C_P or C_V changing with T, use an integral over temperature range.

Degrees of Freedom and Energy Distribution

Molecule consists of atoms connected by covalent bonds. Covalent bonds typically do not break at room temperature.
Molecules and atoms in molecules can move, rotate and vibrate.
Each degree of freedom carries on average energy 1/2 RT / N_Avogadro. 1 mole of 1 degree of freedom carries 1/2 RT
Each vibrational degree of freedom carries an additional 1/2 RT for potential energy. The total energy per one mole of vibrational degrees of freedom is RT.
Effective temperature calculation: Since vibrational energy of one DOF is RT, T_effective = E_mole /R .Example problem: E = 1eV = 100 kJ. What temperature it corresponds to? T = E/R = 11,500K

Calculating Degrees of Freedom of Molecules in Gas Phase

Two general approaches:
1. 3*N_atoms - N_constraints
2. 6 (3 rigid body translations and rotations) + N_{internal_rotations_vibrations}
for a single molecule: one atom: 3 DOF, two atoms (0₂ N₂ etc.) 5 if the bond is NOT excited, 6(7) if excited :

Heat Capacity

Covalent Bond Energies

Bond	Length(Å)	Energy (kJ/mol)
Br-Br	2.28	-193
C-Br	1.94	-288
C-C	1.54	-348
C=C	1.34	-614
C≡C	1.20	-839
C-Cl	1.77	-330
C-F	1.35	-488
C-H	1.09	-413
C-I	2.14	-216
C-N	1.47	-308
C-O	1.43	-360
C-S	1.82	-272
Cl-Cl	1.99	-243
F-F	1.42	-158
H-Br	1.41	-366
H-C	1.09	-413
H-Cl	1.27	-432
H-F	0.92	-568
H-H	0.74	-436
H-I	1.61	-298
H-N	1.01	-391
H-O	0.96	-366
I-I	2.67	-151
N≡N	1.10	-945
N-N	1.45	-170
O-O	1.48	-145
O=O	1.21	-498

Some rules for the bond energies:

bond energies of stable bonds are negative;
the higher the bond order, the lower the bond energy (note that the bond energy is negative, the lower, the more favorable);
we define the bond energy as the energy required to form a bond from elements (opposite sign to what is required to break a bond);
note that some textbooks use the term bond energy as "the bond-breaking energy", rather than "bond-forming" energy;
energy is released when bonds form;
energy is absorbed when bonds break;
many chemistry textbooks mislead students into thinking that "Chemical bonds store energy". Reality is opposite;

Chemical reaction, burning

Balance of bond energies dominates in the heats of chemical reaction
A simple rule:
- heat produced/exothermic_reaction ≡ bonds formed or stronger bonds
- heat absorbed/endothermic_reaction ≡ bonds broken or weaker bonds
Example: 2H2 + O2 => 2H2O : before 3 bonds, after 4 bonds (see table for details). Bonds gained: heat produced.
Example with equal number of bonds but different strength: H2(g) + Br2(g) -> 2 HBr (g). Two for two, but Br2 bond is weak: 2*(-366) -((-436)+(-193)) = -103 (kJ/m): exothermic reaction

Energy comparisons for one photon in electon-volts (eV). 1 eV = 1.6 10^-19 J .

210 MeV: the average energy released in fission of one Pu-239 atom
200 MeV: the average energy released in nuclear fission of one U-235 atom
17.6 MeV: the average energy released in the fusion of deuterium and tritium to form He-4; this is 0.41 PJ per kilogram of product produced
1 MeV (1.602×10⁻¹³ J): about twice the rest energy of an electron
13.6 eV: the energy required to ionize atomic hydrogen; molecular bond energies are on the order of 1 eV to 10 eV per bond
1.6 eV to 3.4 eV: the photon energy of visible light
25 meV: the thermal energy kBT at room temperature; one air molecule has an average kinetic energy 38 meV
230 µeV: the thermal energy kBT of the cosmic microwave background

Math background

Basic facts

e = 2.71828. The number e is an important mathematical constant, approximately equal to 2.71828
ln ≡ log_e
Pythagorean theorem: in a right-angled triangle with sides a,b, and hypotenuse c : a² + b² = c²
Area of a circle is πr² where r = radius of a circle
Volume of a cylinder is πr²h where r = radius of circular face, h = height
Volume of a sphere : (4/3)πr³

Weighted average

Simple average: A, B, C (N=3) Average = ( A + B + C )/N

Weighted average: A,B,C, W_A W_B W_C .

Weighted_Average = (A*W_A + B*W_B + C*W_C )/ ( W_A + W_B + W_C )

Trigonometry

Use the trigonometric circle. Note that angles are measured in radians or degrees ° ( e.g. π/2 is 90° , π is 180° etc.). DO NOT TRY TO use calculator for a trigonometry question to calculate cos( π/ 2 ) , - the calculator will give you an answer as if it is the number of degrees. The degree is usually explicitly stated with the ° symbol, the presence of π in the formula means that the value is in radian units.

sin( 0 ) = sin( π ) = sin( 2π )=0	cos( 0 ) = cos ( 2π) = 1.; cos( π ) = -1
sin( ½π ) = 1.	cos( ±½π ) = 0.
sin( -α ) = - sin( α )	cos( -α ) = cos( α )

Logarithms

Definition: log_a (x) = y means x = a^y

formula example
log(xy) = log(x) + log(y)
log(x/y) = log(x) - log(y)
log(1/y) = -log(y) log₁₀ (0.01) = log₁₀ (1/100) = -log₁₀ (100) = -2
log(x^p ) = p log(x) log₁₀ 1000 = log₁₀ 10³ = 3 log₁₀ 10 = 3
ln(x)≡log_e (x) = 2.3 log₁₀ (x)

formula	example
log(xy) = log(x) + log(y)
log(x/y) = log(x) - log(y)
log(1/y) = -log(y)	log₁₀ (0.01) = log₁₀ (1/100) = -log₁₀ (100) = -2
log(x^p ) = p log(x)	log₁₀ 1000 = log₁₀ 10³ = 3 log₁₀ 10 = 3
ln(x)≡log_e (x) = 2.3 log₁₀ (x)

Derivatives

Notations

(rarely used Newton's notation is omitted):

df/dx , d² f/dx²	Leibniz	1st and 2nd derivative
f'(x), f''(x)	Lagrange	1st and 2nd derivative
∂f(x,y,z..)/∂x	Leibniz	partial derivative

Differentiation rules

Type	Function	Derivative	Example
Constant	f(x) = a	f'(x) = 0	a' = 0
Power	f(x) = xⁿ	f'(x) = nx^n-1	(x² )' = 2x
Product	f • g	f' g + f g'
Quotient	f / g	( f' g - f g' )/( g² )
Chain	f(g(x))	(df/dg)(dg/dx)
Exponent	e^x	e^x
Exponent	e^ax	ae^x
Exponent	a^x	a^x • log(a)
Log	ln(x)	1/x
Sin	sin(x)	cos(x)
Cos	cos(x)	-sin(x)

Integrals

Type	Function	Integral
Constant	a	ax + C
Power	xⁿ	x_n+1 /(n+1) + C
One over x	x^-1	ln(x) + C (x>0)
Exponential	a^x	a^x /ln(a) + C

Unit conversions and main constants

Celsius	Kelvin - 273.15
Kelvin (K)	Celsius + 273.15
1 kJ	0.239 kcal
1 kcal	4.184 kJ
1 eV	1.6 × 10^-19 joules
1 L (liter)	10^-3 m³
1 meter/second	2.237 Miles/hour
1 Mile/hour	0.44704 meter/second
1 mole of sub.	MW gram (MW is in Daltons, of Da).
1 g of sub.	1/MW moles

Physical Constants

abbrev.	definition	exact value, units	approximation
c	speed of light	299 792 458 m / s	3 10⁸
G	Gravitational constant	6.673×10⁻¹¹ N·(m/kg)2
g	acceleration of free fall	9.8 m/s²
h	Planck constant	6.6260695 ×10⁻³⁴ J·s	6.626×10⁻³⁴
k_e	Coulomb's constant	8.98755×109 N·m² /C²
N_A	Avagadro constant	6.02214129×10²³ mol⁻¹	6×10²³
R	gas constant	8.3144621 JK^-1 mol^-1	8.3

Glossary

Root mean square value (rms)

If one has an array of N values of X_i, e.g. velocities of many molecules, the root mean square value can be calculated by the following process: (i) average the squared values, (ii) take a square root of the average:

X_rms = (( ∑ X²_i )/ N )^½

Physical Chemistry of Drugs. Summary

Objectives

Drugs

Drugs by state of matter or phase

Units, scales, force, energy, temperature

Moving Molecules, Gas Law

Thermodynamics

Heat Capacity

Degrees of Freedom and Energy Distribution

Calculating Degrees of Freedom of Molecules in Gas Phase

Heat Capacity

Covalent Bond Energies

Math background

Basic facts

Weighted average

Trigonometry

Logarithms

Derivatives

Notations

Differentiation rules

Integrals

Unit conversions and main constants

Physical Constants

Glossary

Root mean square value (rms)