Advanced Molecular Modelling Applied to Drug Discovery(3FK181)，Uppsala University Exam Preparation Notes- 2023

Note

This is an personal examination preparation note for the course Advanced Molecular Modelling Applied to Drug Discovery(3FK181)，Uppsala University.Converted atuomatically by Claude AI

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Chapter 1: Ligand-Protein Interaction

Key Points:

1. Describing the interaction of molecules

   • Determining if a molecule is a good target or not

2. Types of Interactions:

   • Electrostatics: Follow Coulomb’s Law

     • F = -q₁q₂/4πε₀r² (ε₀: electric constant, q₁, q₂: charges)
     • π-π stacking: interactions between benzene rings
     • π-cation interactions: e.g., AchE with -NH₃⁺ and -Ph groups

   • H-bond: Consider O, N atoms

     • Mention HBD (hydrogen bond donors): -OH, -NH₂
     • HBA (hydrogen bond acceptors): C=O, C≡N (有供氢的)

   • Halogen bond: Cl···O, Br···O, I···O

     • No F bonds; if there is a -OF₃ group, it shows hydrophobic effect

   • Polarization: Charge redistribution when molecules approach each other

     • Charge → Dipole
     • Dipole → Dipole
     • Dipole-Dipole Interaction → Polarization → London Dispersion
     • Not generally included in molecular mechanics

   • London Dispersion: Dispersive forces (色散力)

     • π-stacking interactions

3. Covalent Bond:

   • Advantages:
     • Potency
     • Selectivity
     • PD (Pharmacodynamics)
   • Risks: Toxicity
   • Explanation:
     • Potency: Interaction is irreversible, very stable
     • Selectivity:
       • Can interact with specific proteins
       • Afatinib: Irreversibly binds certain proteins
     • PD: Not requiring high blood concentration; takes a long time
     • Toxicity: Off-target effects, irreversible

4. Hydrophobic effect: An entropic effect (熵的)

   • Gains potency but reduces solubility
   • Order of the dynamic hydrogen bond in water to accommodate a non-polar solute

5. Considerations for L-P (Ligand-Protein) interaction:

   • Size
   • Shape
   • Exposure (暴露外表的景象)
   • Hydrophilic/Hydrophobic properties
   • Ligandability/Druggability
   • Flexibility

6. Relative Strength of Interactions:

   • Covalent > Electrostatic > H-bond > van der Waals
   • 力的作用力大小

Chapter 2: Molecular Dynamic Simulation

I. Why Simulation

1. Study larger biological systems
2. Calculate movement as a function of time
3. Predict and explain outcomes of biological experiments
   • Example: Butane (丁烷) conformational analysis → Peptide folding

II. Molecular Mechanics Approximation

1. Ignore electron motion
2. Born-Oppenheimer approximation: Separate electron and nuclear movements
3. Use Newton’s laws to describe the system
4. Apply small molecule parameters to larger systems

III. Van der Waals Force

• Repulsive then attractive
• UVdW = 4ε[(σ/r)¹² - (σ/r)⁶]
  • σ: Collision diameter
  • ε: Well depth
• Non-bonded interaction

IV. For Charged Molecules

• UC = q₁q₂/4πε₀r₁j
• Very expensive to calculation

V. Bond Potential (Hooke’s Law)

• Accurate Morse potential typically replaced with simpler harmonic function (简谐函数)
• Bond: U = k₁/2(r-r₀)²
• Angle: U = k₂/2(θ-θ₀)²

VI. Consider Partial Charges

• Water: partially charged → Water models: TIP3P, SPC

VII. Consider Torsional Potential (扭转势能)

• U(φ) = Σ(k_n/2)[1+cos(nφ-φ₀)]

  • n = number of minima
  • 若有3个 minimal, n=3, φ₀=0
  • U(φ) = k₁/2[1+cos(φ)] + k₂/2[1+cos(2φ)] + k₃/2[1+cos(3φ)]

• Problems:

  1. Complex shape
  2. Relatively weak
  3. Torsion may lead to large changes to the whole system
  • Addition: Improper dihedrals → keep 4 atoms in a plane

• Total energy function: U = 4ε[(σ/r)¹² - (σ/r)⁶] + q₁q₂/4πε₀r₁j + k₁/2(r-r₀)² + k₂/2(θ-θ₀)² + Σ(k_n/2)[1+cos(nφ-φ₀)] + k₃/2(ξ-ξ₀)²

Summary of Force Field

• It’s empirical! (经验上)

1. There is no correct way to derive a force field; experimental data improves it
2. Fits reality, not theory - aim is to reproduce experimental results, not calculate it
3. Predict, not explain
4. Consider computational time; the best model isn’t always chosen

Biomolecular Simulation

VIII. AMBER

• Assisted model building with energy refinements
• Helps with:
  • Bond stretching, angle bending, dihedral torsional angles
  • Interaction with other models
  • Structure

Force Field Summary

1. Force field is simplified and empirical (经验上)
2. Form and parameters differ in different models
3. Usefulness depends on the question being asked
4. Major problem: Bonds cannot be broken
5. Transferability of parameters is unclear
   • The ability to transfer parameters from one molecule to another (参数的转移性)
   • Assumption: similar molecules have the same parameters
   • Even if they have similar structure
   • Consider environment, temperature, and other molecules
6. Polarization is not included
7. General methods of generating parameters are still under development

Modeling Complex Biomolecules

• Formula combines bond energies, angle energies, torsional terms, improper terms, electrostatic and VDW terms
• We have force field parameters for amino acids
• What about larger biomolecules?

Free Energy vs Potential Energy

• The minimum potential energy state need not be the most popular state; free energy is
• G = H - TS → Entropy
• Enthalpy considerations
• Take average of a huge number of particles
• Experiment observation based on a single configuration

Time Average

• < A > = (1/τ)∫₀ᵗ A(r^N(t), p^N(t))dt
• r: position; p: momenta; of all particles N
• (粒子位置和动量), τ→∞, 平均即为实际

Statistical Thermodynamics

• Know the probability of every state
• n_i = N·(e^(-βE_i)/Σⱼe^(-βE_j))
  • n_i: number in a state
  • 利用"概率"
• Ergodic hypothesis (遍历假设): Time average = Ensemble average

Sampling an Energy Landscape

• MD Simulation uses:
  • Newton’s laws
  • Time dependence
  • Time average
  • Dynamical events

Molecular Dynamics Simulation (分子动力学 Force Field)

1. Try to calculate the path of individual particles
2. Errors occur but final results are acceptable
3. Calculate motion as a function of time

MD Process

1. Start with a set of coordinates
2. Calculate potential and force (用时时所迭)
3. Calculate acceleration: a_i = F_i/m_i
4. Calculate velocity and position: v + dt

Leap-Frog Algorithm

• Calculate new coordinates for atoms (迭代算法)
• r_i(t+dt) = r_i(t) + dt·v_i(t+dt/2)

For Trajectory (轨迹)

1. Different time series of snapshots
2. In theory, MD simulation is deterministic (确定性的)
   • If we know the state, we can predict behavior
3. In practice, errors accumulate
4. Looking over long time provides a good picture

How to Choose Time Step

1. Must be small but not too small (need more to reach 1ms simulation)
2. If step is too big → unreasonable positions
3. Determined by the fastest vibration in system
4. Typically choose 1-2 fs

Solvation in MD

1. Vacuum: No solvent, problematic

2. Implicit solvent model:

   • Replaced by average behavior of water
   • Analytical, so calculations are fast
   • Disadvantage: Only electrostatics in this expression
   • Other effects like entropy must be added separately

3. SBC (Spherical Boundary Condition):

   • Add water near the place of interest
   • Save time if not considering whole protein
   • Should restrain water inside the spherical boundary
   • Should add right polarization

4. PBC (Periodic Boundary Condition):

   • Create “infinite” system by creating identical copies
   • Use a periodic cell shape and cut-offs
   • When we define a “primary cell”, we copy it
   • When an atom reaches edge, it enters from another side
   • Advantage: Use less atoms to simulate a large system
   • Computationally expensive

Common Ensembles

1. NVT (Canonical): Control T (等容等温)
2. NPT (Isobaric-Isothermal): Control T & P (等压等温)
   • Temperature → Kinetic energy → Velocity

Limitations of MD

1. Parameters are not perfect
   • Amino acid solvent free energy: ~1kJ/mol
   • Impossible to calculate binding energy more accurately
2. Phase space not fully sampled (相空间未完全采样)
3. Limited polarization effects (极化限制):
   • Water reorientation
   • Partial charge is fixed (The orientation change)

MD Workflow

Get structure → Fix structure → Prepare topology → Add water → Energy min. → Equilibration phase → Run → Analysis

MD Outcomes

• Doesn’t mean it’s “real”
• Everything is about statistics
• Error? A signal event is not

Chapter 3: Free Energy

I. Free Energy Basics

• ΔG<0: Reaction is spontaneous (自发的)

II. Absolute Free Energy Calculation

• Hard/impossible to calculate directly
• Need to sample all parts of phase space
• Typical simulations only sample low-energy regions

III. Thermodynamic Perturbation (干扰)

• F_A→B = F_B - F_A = -k_B·T·ln(Q_B/Q_A)
• Zwanzig’s Formula: F_A→B = -kT·ln<e^(-ΔH/β)>_A
• How to use Zwanzig’s Formula:
  • Calculate ΔG to go to B
  • Potential Energy of A and B (势能)
• Assumption: A-B gap is not big

IV. Thermodynamic Cycle

• Path not necessary to be meaningful
• A → B → C → D: ΔG_AB + ΔG_DA - ΔG_CB - ΔG_DC = 0

V. Fixing the Overlap Problem

• H(λ) = (1-λ)H_A + λH_B
  • λ is interpolation parameter (加入λ为中间态)
• F_B - F_A = (F_λ=0.5 - F_λ=0) + (F_λ=1 - F_λ=0.5)

VI. Solvation Process

• ΔG_Sol = ΔG_VdW + ΔG_Charge + ΔG_Cav
  • VdW: van der Waals interaction between L-J atoms
  • Charge: Electrostatics
  • Cav: Cost for creating cavity in water (surface/entropy)

VII. Thermodynamic Integration (积分)

• ΔF = ∫₀¹<H_B - H_A>_λ dλ

VIII. Inhibitor Design

• If we know an inhibitor, we want a similar one (I = protein)
• 用于创新设计算法

1
2
3

I+E → I'+E'
↑     ↑
I     I'

• F_I→I’_bind - F_I→I’_bind = F_I→I’_bound - F_I→I’_Free

Chapter 4: Quantum Chemistry

I. Why Quantum Chemistry?

1. No parameters required
2. Explicit treatment of electrons
   • In principle, everything can be modeled:
   • Polarization, reactions, spectroscopy

II. Postulates of Quantum Mechanics

1. State of quantum mechanical system completely specified by wavefunction Ψ(r,t)

   • r: vector in 3D space (三维空间的一个方向量)
   • t: time
   • Born Interpretation: Square of wave function → probability of finding the system at given values of the variable
   • p(x₁,x₂) = ∫|Ψ(x)|²dx
   • 概率

2. For every observable in classical mechanics, there is an operator in QC

   • (把传统力学量用量子算符表达互联系起来)

3. In any measurement of the observable associated with operator Â, the only values ever observed are the eigenvalues a

   • ÂΨ = aΨ
   • Â为物理量, Â对应于ψ, 则Â的a作用于ψ, 则Âψa

4. It can be averaged:

   • < A > = ∫ΨÂΨdτ/∫ΨΨdτ

5. Schrödinger Equation:

   • Time-dependent: ÂΨ(r,t) = iℏ∂Ψ(r,t)/∂t
   • For time-independent systems: (Â-E)Ψ(r) → the system’s eigenvalues

Born-Oppenheimer Approximation

1. Assumes m_nuclei » m_electron

   • Electrons move faster
   • Electrons adjust immediately to nuclear movement
   • The operator of electrons doesn’t include the kinetic of nuclei (动核近似)
   • H_e = T_e + V_ee + V_en + V_nn

2. Separation of Variables

   • n electrons moving independently: Ψ = ΠΦᵢ
   • E = ΣEᵢ

3. Pauli Principle and Slater Determinant

   • Pauli exclusion principle: Two or more electrons cannot occupy the same quantum state
   • (一个电子不可处于同一量子态)
   • Slater determinant: Follow the Pauli principle
   • When two electrons exchange location, wavefunction changes sign
   • For n=2 electrons: Ψ = (1/√2)|Φ₁(1) Φ₂(1)| / |Φ₁(2) Φ₂(2)|
   • Consequences of Slater determinant:
     • Follow electron Pauli principle
     • Consider electron correlation
     • Consider charge exchange

Disadvantages

• Not always accurate
• Cannot explain correlation energy
• Computationally difficult

III. Hartree-Fock / Self-Consistent Field

1. Slater determinant provides mathematical framework → Hartree-Fock

2. Hartree-Fock calculates the energy of a system (能量)

3. More accurate than simpler methods (更为准确)

4. Hartree-Fock uses basis sets to construct wavefunctions, then adjusts to reality

5. To choose basis sets:

   • Atomic nuclei as centers
   • Mix and match these functions
   • Base: Solution of Schrödinger equation of H
   • Radial part is Slater-type orbital (STO) - an exponential function
   • Use Gaussian functions (GTOs) to approximate

6. How many basis sets?

   • One independent minimal set
   • For reasonable results: at least 2 or 3
   • Add flexibility: add polarization
   • For anions: diffuse functions

7. Split Valence Basis Sets (价层分裂):

   • Valence electrons are more important
   • Use more functions on valence electrons
   • 6-31G(d,p) notation:
     • 6: inner shell electrons represented by 6 Gaussian-type orbitals
     • 3,1: valence shell represented by 3 GTOs for first part, 1 GTO for second
     • d,p: polarization functions (d for heavy atoms, p for hydrogen)

8. Limits of Hartree-Fock:

   • Adding more basis sets is always better
   • In H-F theory, electron-electron interactions treated with mean-field approach (平均场)
   • In reality, electron movement is correlated
   • Missing: Electron correlation energy (能量) 相关性

9. Post-Hartree-Fock:

   • Correlation effects approximated by including additional determinants
   • CI, MCSCF, MPn methods available

IV. Density Functional Theory

1. Energy is a function of electron density:

   • E = F[ρ(r)] where ρ is density, F is a functional
   • Don’t need to solve Schrödinger function
   • Only 3N variables needed
   • Hohenberg-Kohn theorem states the function “EXISTS” but doesn’t specify what it is

2. We know it includes these parts:

   • E[ρ] = T[ρ] + E_ne[ρ] + E_ee[ρ]
     • T[ρ]: Kinetic energy of electrons
     • E_ne[ρ]: Attraction between nuclei and electrons
     • E_ee[ρ]: Repulsion between electrons
   • E[ρ] = T[ρ] + E_ne[ρ] + J[ρ] + K[ρ]
     • J[ρ]: Coulomb interaction (库仑项)
     • K[ρ]: Exchange term to fit Pauli exclusion principle (交换项)

3. Kohn-Sham Approach:

   • According to Heisenberg uncertainty principle, we cannot test momentum and location at the same time
   • (测不准原理不确定性原理)
   • P为运动量，对应位置，教授动能
   • Need to know velocity → 方程式
   • Hartree-Fock can calculate kinetic energy
   • For Kohn-Sham approach in DFT, add “fake” electrons to get approximate kinetic values

4. DFT in Practice:

   • E[ρ] = T_s[ρ] + E_ne[ρ] + J[ρ] + E_xc[ρ]
   • T_s[ρ]: Kinetic energy from Kohn-Sham orbitals
   • E_xc[ρ]: Other quantum mechanical effects (correlation)
   • Kohn-Sham simplified the function to T_s[ρ] + E_xc[ρ]
   • E_xc[ρ] is hard to calculate
   • Common functionals: B3LYP, 6-31G
   • Balance quality and computational cost

5. Dispersion in DFT:

   • Though dispersion is weak, it increases quickly in larger systems
   • Short distance: electron correlation included in functionals
   • Long distance not included
   • Most functionals fail to represent dispersion at all distances
   • Modern functionals (Minnesota family) take this into account

6. DFT in Drug Design:

   • Parameter generation for small molecules

7. Semi-Empirical Method (半经验方法):

   • These methods require too much computational time
   • Strategies to reduce computation:
     • Reduce number of functions
     • Reduce number of integrals (积分)
   • Approximations made to Hartree-Fock:
     • Core electrons ignored; only consider valence electrons with effective charge Z_eff
     • Only minimal basis set employed
     • Integrals involving more than two centers neglected
     • Remaining integrals are parameterized

8. Neglect Diatomic Differential Overlap approximation:

   • Examples: PM3, PM6, PM7

V. Applying QC (Quantum Chemistry)

1. Energy Landscape:

   • QC bonds are a consequence of PES, not input parameters
   • Different molecules can be part of the same PES

2. Transition State Theory:

   • Reaction coordinate: lowest pathway from reactant to product
   • Transit state: max energy of reaction coordinate
   • This state/enzyme must be stable

3. Geometry Optimization:

   • For force fields, minimum energy when gradient becomes close to zero (energy minimum)
   • For QC:
     • Several maxima/minima
     • Optimization to saddle points
   • Curvature calculation needed (曲率)
   • Hessian Matrix: Contains all second derivatives
     • For energy minimum, all eigenvalues are positive
     • For transition state, one eigenvalue is negative
     • For second-order saddle point, two eigenvalues are negative / opposite sign

4. Hessian Matrix and Spectroscopy:

   • Hessian matrix used to understand vibration
   • If molecular vibration is harmonic, we can solve Schrödinger equation
   • E_vib = ℏ(n+½)√(k/μ), n = energy level
   • Between different energy levels → spectroscopic transition

5. Isotope Effect (同位素):

   • Start from E⁰_vib = ½ℏ√(k/μ) where μ is reduced mass
   • For H and D as example:
     • E⁰_H > E⁰_D, so ΔE_H < ΔE_D
     • In drug design: -O-CH₃ → -O-CD₃
     • Metabolized slowly, less frequent dosing

6. Enthalpies and Free Energy:

   • QC energy results include:
     • Electrons at 0K
     • Electrons and nuclei strictly separated
   • Absolute energy not useful; compare to others with same electron and nuclei configurations, using same method and basis set
   • Electronic energy → Enthalpies considering thermal energies (translation, rotation, vibration)

Chapter 5: Cheminformatics

I. Different Databases

Database comparison:

II. Cheminfo Selection

1. Pre-clean the Database:

   • Remove wrong structures
   • Remove salt and inorganic compounds
   • Remove duplicated and tautomers
   • Manual inspection, remove obvious unreasonable data

2. General Chemical Prototyping:

   • Compare active compounds if available
   • Exclude toxic structures and structures leading to similar analogues
   • Consider hydrophobic cores, rotatable bonds if target is known
   • Remove reactive functional groups (ensure selectivity)
   • Remove PAINS (Pan-Assay Interference Compounds) to ensure druggability
   • Create analogues to test affinity with docking

3. Diversity: PCA with physical chemical parameters, clustering

4. If high affinity is found, use it as a lead compound considering:

   • Size
   • Flexibility
   • Lipophilicity

Chapter 6: Virtual Screening

I. Virtual Screen vs. High-throughput Screening

Comparison:

II. Virtual Screen Methods

• Docking: Glide
• Pharmacophore search: Phase
• Shape similarity: ROCS
• Electrostatic similarity: EON

III. Ligand Efficiency

• LE = ΔG/N_non-H, where ΔG = -RTlnKd

1. Smaller size means higher affinity to dock the target
2. High efficiency means more druggability, more stable
3. Normalized potency of a hit

IV. Negative and Positive Design

• Negative Design: Avoid undesired properties (e.g., toxicity)
• Positive Design: Attempt to engineer molecules with desired properties

Process:

1. Weeding Out (Negative design):

   • Drug likeness
   • In silico ADMET
   • Frequent hitters
   • Reactive groups

2. Narrowing Down (Positive design):

   • Trend vectors
   • Substructural analysis
   • Similarity searching
   • Profile/landscape analysis (size, electrostatics, HBD/HBA)

3. Focusing In:

   • Structural based models
   • Advanced pharmacophore models
   • Informed docking
   • Informed design

V. Frequent Hitters and Promiscuous Binders

1. Frequent hitters: Show activity in many assays; unspecific binding, aggregate, auto-fluorescence. May not be bioactive.
2. Promiscuous binders: Subset of frequent hitter. Can interact with many targets. Bioactive!

VI. Benchmark Data Sets

1. Decoy: Has similar chemical structure as active compounds but doesn’t have chemical activity. Use as negative controls in VS.

2. DUD: Directory of decoys, BRAD for ligand-base

3. MUV (Maximum Unbiased Validation): Collection of benchmark datasets

   • DUD: OK for docking
   • MUV: If we want minimal bias

4. Bias: May be easy to select similar molecules but not normal ones

5. Process to create own database benchmark:

   1. Create a goal/purpose
   2. Collect active compounds
   3. Collect inactive compounds (ZINC Database → “drug-like”)
   4. Ensure diversity with different structures
   5. Split data into training and validation sets
   6. Test the method

VII. Data Fusion

• Sum rank, rank vote, sum score
• EF = (tp/(tp+fp))/(A/T)
• Pareto rank
• Test EF 1%, EF 10%

VIII. Problems with Decoys

1. Random compounds, decoys data should be similar, cannot divide the active compounds
2. Too similar: Hard to divide and may cause bias
3. From DUD: Some inactive may have active to other target

MD & QC Summary

I. Why MD vs Why QC

MD Purpose:

• Study biological systems
• Calculate with function of time
• Predict and explain outcomes of biological experiments

QC Purpose:

• No additional parameters required
• In theory, everything can be calculated

II. MD Approximation vs QC Postulates

MD Approximations:

• Ignore electron motion
• Separate electron and nuclei
• Use Newton’s law
• Transfer parameters from small to large molecules

QC Postulates:

• Wavefunction Ψ(r,t), Born’s interpretation (解释)
• For every observation, an operator
• For every operator, eigenvalues
• Schrödinger’s equation

III. Force Field Advantages/Disadvantages

1. Force Field is simplified and empirical (经验上)
2. Form and parameters differ in different models
3. Usefulness depends on questions asked
4. Major problem: bonds cannot be broken
5. Transferability of parameters unclear
6. Polarization not considered
7. General methods still developing

IV-V. MD Simulation Process

1. Try to calculate the path of each individual
2. Newton’s law
3. May have errors
4. Add-up errors may be OK
5. Calculate motion as function of time

VI. Limits of MD

1. Parameters not perfect
2. Phase space not fully sampled
3. Limited polarization effects (极化效果)

VII. QC Approaches

• Schrödinger Equation: H(r,t) = EΨ(r,t)
• Pauli exclusion principle and Slater determinant
• With Slater determinant + Basis Set → Hartree-Fock
• Using GTO, STO-nG models (STO-3G means STO approximated by 3 GTOs)
• Hartree-Fock uses mean-field approach, but electrons’ interaction is correlated
• Post Hartree-Fock: CI, MPn
• Density Functional Theory: No need to solve Schrödinger, only 3N
• DFT = E[ρ] = T_S[ρ] + E_ne[ρ] + J[ρ] + E_xc[ρ]
• DFT principle accurate; HF is not
• But we still don’t know the expression of E_xc[ρ]
• Choose B3LYP/6-31G to balance quality and cost
• Semi-Empirical Method: Reduce number of functions/integrals
• Model PMx (x=3,6,7)
• Hessian Matrix: Contains all second derivatives
• For energy min: all eigenvalues positive
• For TS: one eigenvalue negative
• A second-order saddle point: two eigenvalues negative
• Also in spectroscopy: E_vib = ℏ(n+½)√(k/μ)
• Isotope effect

Words List

• Electrostatic: 电力
• π-stacking
• Halogen Bond
• London Dispersion: a 色散力
• Pauli Repulsion: 泡利(排斥)
• Entropic: 熵的
• Enclosure/Exposure: 暴露外表的景象
• Torsional Potential: 扭转势能
• Ensemble Average: 总体~
• Ergodic hypothesis: 遍历假设
• Ensemble average = Time average
• Trajectory: 轨迹
• Implicit solvent model
• Canonical Ensembles: 等温
• Isobaric Ensemble
• Postulate: 假设
• Operator: 算子
• Eigenvalue: 特征值
• Spherical/Periodic