{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Boltzmann Populations and Conformational Analysis\n", "\n", "This notebook covers the statistical mechanics of conformer distributions, essential for:\n", "\n", "- **NMR analysis** - Boltzmann-averaged coupling constants and NOEs\n", "- **Drug binding** - accessible conformations for receptor interaction\n", "- **Property prediction** - ensemble-averaged molecular properties\n", "- **Free energy calculations** - conformational entropy contributions\n", "\n", "## Statistical Mechanics Background\n", "\n", "### Boltzmann Distribution\n", "\n", "At thermal equilibrium, the population of conformer $i$ is:\n", "\n", "$$p_i = \\frac{e^{-E_i/k_BT}}{\\sum_j e^{-E_j/k_BT}} = \\frac{e^{-E_i/k_BT}}{Z}$$\n", "\n", "where $Z$ is the partition function.\n", "\n", "### Useful Numbers at 298K\n", "\n", "| ΔE (kcal/mol) | Population Ratio |\n", "|---------------|------------------|\n", "| 0.0 | 1.0 : 1.0 |\n", "| 0.6 | 2.7 : 1.0 |\n", "| 1.0 | 5.4 : 1.0 |\n", "| 1.36 | 10 : 1 |\n", "| 2.0 | 29 : 1 |\n", "| 2.72 | 100 : 1 |\n", "\n", "Rule of thumb: **1.36 kcal/mol = 10:1 ratio at room temperature**" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "import os\n", "import tempfile\n", "from pathlib import Path\n", "\n", "import numpy as np\n", "import Auto3D\n", "from Auto3D import Auto3DOptions, main\n", "from rdkit import Chem\n", "from rdkit.Chem import AllChem, Descriptors, rdMolDescriptors\n", "from rdkit.Chem import TorsionFingerprints\n", "import pandas as pd\n", "\n", "print(f\"Auto3D version: {Auto3D.__version__}\")\n", "\n", "# Constants\n", "R = 1.987e-3 # kcal/(mol·K)\n", "T = 298.15 # K\n", "RT = R * T # ~0.593 kcal/mol at 298K" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 1. Basic Boltzmann Analysis\n", "\n", "Let's analyze conformer populations for a flexible drug molecule." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "def calculate_boltzmann_populations(energies_kcal, T=298.15):\n", " \"\"\"\n", " Calculate Boltzmann populations from conformer energies.\n", " \n", " Args:\n", " energies_kcal: Array of conformer energies in kcal/mol\n", " T: Temperature in Kelvin\n", " \n", " Returns:\n", " Array of populations (sum to 1.0)\n", " \"\"\"\n", " energies = np.array(energies_kcal)\n", " RT = R * T\n", " \n", " # Shift to prevent numerical overflow\n", " energies_shifted = energies - energies.min()\n", " \n", " # Boltzmann weights\n", " weights = np.exp(-energies_shifted / RT)\n", " \n", " # Normalize to populations\n", " populations = weights / weights.sum()\n", " \n", " return populations\n", "\n", "\n", "def calculate_conformational_entropy(populations):\n", " \"\"\"\n", " Calculate conformational entropy from populations.\n", " \n", " S_conf = -R * Σ p_i * ln(p_i)\n", " \n", " Returns entropy in cal/(mol·K)\n", " \"\"\"\n", " # Filter out zero populations\n", " p = populations[populations > 0]\n", " S_conf = -R * 1000 * np.sum(p * np.log(p)) # Convert to cal/(mol·K)\n", " return S_conf\n", "\n", "\n", "# Example: hypothetical conformer energies\n", "example_energies = [0.0, 0.5, 1.0, 1.5, 2.0, 3.0, 4.0, 5.0]\n", "\n", "pops = calculate_boltzmann_populations(example_energies)\n", "S_conf = calculate_conformational_entropy(pops)\n", "\n", "print(\"Boltzmann Population Analysis:\")\n", "print(\"-\" * 50)\n", "for i, (e, p) in enumerate(zip(example_energies, pops)):\n", " bar = \"█\" * int(p * 50)\n", " print(f\"Conf {i+1}: E={e:4.1f} kcal/mol, pop={p*100:5.1f}% {bar}\")\n", "print(\"-\" * 50)\n", "print(f\"Conformational entropy: {S_conf:.2f} cal/(mol·K)\")\n", "print(f\"-T*S_conf at 298K: {-T * S_conf / 1000:.2f} kcal/mol\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 2. Conformational Analysis of a Drug Molecule\n", "\n", "Let's analyze a real drug molecule with Auto3D." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# Metoclopramide - a flexible drug with multiple rotatable bonds\n", "# Used as antiemetic\n", "drug_smiles = {\n", " \"metoclopramide\": \"CCN(CC)CCNC(=O)C1=CC(=C(C=C1OC)N)Cl\",\n", "}\n", "\n", "# Count rotatable bonds\n", "mol = Chem.MolFromSmiles(drug_smiles[\"metoclopramide\"])\n", "n_rot = Descriptors.NumRotatableBonds(mol)\n", "\n", "print(f\"Drug: Metoclopramide\")\n", "print(f\"Rotatable bonds: {n_rot}\")\n", "print(f\"Theoretical max conformers: ~{3**n_rot} (3 states per bond)\")" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# Write to file\n", "with tempfile.NamedTemporaryFile(mode='w', suffix='.smi', delete=False) as f:\n", " for name, smi in drug_smiles.items():\n", " f.write(f\"{smi} {name}\\n\")\n", " drug_file = f.name\n", "\n", "print(f\"Input file: {drug_file}\")" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# Generate multiple conformers with Auto3D\n", "# Use window mode to get all conformers within energy range\n", "\n", "if __name__ == \"__main__\":\n", " config = Auto3DOptions(\n", " path=drug_file,\n", " window=5.0, # All conformers within 5 kcal/mol\n", " optimizing_engine=\"AIMNET\",\n", " threshold=0.5, # RMSD threshold for diversity\n", " max_confs=300, # Generate enough initial conformers\n", " use_gpu=True,\n", " )\n", " \n", " conformer_output = main(config)\n", " print(f\"Output: {conformer_output}\")" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "def analyze_conformer_ensemble(sdf_path):\n", " \"\"\"\n", " Comprehensive analysis of conformer ensemble.\n", " \"\"\"\n", " mols = list(Chem.SDMolSupplier(sdf_path, removeHs=False))\n", " \n", " # Extract energies\n", " energies = []\n", " for mol in mols:\n", " if mol is None:\n", " continue\n", " if mol.HasProp(\"E_rel\"):\n", " e = float(mol.GetProp(\"E_rel\"))\n", " elif mol.HasProp(\"E_hartree\"):\n", " e = float(mol.GetProp(\"E_hartree\")) * 627.509\n", " else:\n", " continue\n", " energies.append(e)\n", " \n", " energies = np.array(energies)\n", " \n", " # Make relative\n", " if energies.min() > 0: # Already relative\n", " energies_rel = energies\n", " else:\n", " energies_rel = energies - energies.min()\n", " \n", " # Calculate populations\n", " populations = calculate_boltzmann_populations(energies_rel)\n", " \n", " # Statistics\n", " S_conf = calculate_conformational_entropy(populations)\n", " \n", " # Effective number of conformers\n", " # (Perplexity - how many conformers are \"effectively\" populated)\n", " effective_n = np.exp(-np.sum(populations * np.log(populations + 1e-10)))\n", " \n", " return {\n", " \"n_conformers\": len(energies),\n", " \"energies_rel\": energies_rel,\n", " \"populations\": populations,\n", " \"S_conf\": S_conf,\n", " \"effective_n\": effective_n,\n", " \"e_range\": energies_rel.max() - energies_rel.min(),\n", " \"mols\": mols\n", " }\n", "\n", "\n", "if 'conformer_output' in dir():\n", " analysis = analyze_conformer_ensemble(conformer_output)\n", " \n", " print(f\"Conformer Ensemble Analysis:\")\n", " print(\"=\" * 50)\n", " print(f\"Total conformers: {analysis['n_conformers']}\")\n", " print(f\"Energy range: {analysis['e_range']:.2f} kcal/mol\")\n", " print(f\"Conformational entropy: {analysis['S_conf']:.2f} cal/(mol·K)\")\n", " print(f\"Effective conformers: {analysis['effective_n']:.1f}\")\n", " print(\"=\" * 50)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# Show top conformers\n", "if 'analysis' in dir():\n", " print(\"\\nTop 10 Conformers by Population:\")\n", " print(\"-\" * 50)\n", " \n", " # Sort by population\n", " sorted_idx = np.argsort(analysis['populations'])[::-1]\n", " \n", " cumulative = 0\n", " for rank, idx in enumerate(sorted_idx[:10]):\n", " e = analysis['energies_rel'][idx]\n", " p = analysis['populations'][idx]\n", " cumulative += p\n", " bar = \"█\" * int(p * 50)\n", " print(f\"#{rank+1:2d}: E={e:5.2f} kcal/mol, pop={p*100:5.1f}%, cum={cumulative*100:5.1f}% {bar}\")\n", " \n", " print(\"-\" * 50)\n", " print(f\"Top 10 conformers account for {cumulative*100:.1f}% of population\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 3. Boltzmann-Weighted Properties\n", "\n", "For comparison with experimental observables, properties should be Boltzmann-averaged." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "def boltzmann_average_property(values, populations):\n", " \"\"\"\n", " Calculate Boltzmann-weighted average of a property.\n", " \n", "

= Σ p_i * P_i\n", " \"\"\"\n", " return np.sum(values * populations)\n", "\n", "\n", "def calculate_conformer_properties(mols, populations):\n", " \"\"\"\n", " Calculate Boltzmann-averaged molecular properties.\n", " \"\"\"\n", " # Calculate properties for each conformer\n", " dipoles = [] # Would need QM for accurate dipoles\n", " radii_gyration = []\n", " asphericity = []\n", " \n", " for mol in mols:\n", " if mol is None:\n", " continue\n", " \n", " # Radius of gyration\n", " try:\n", " rg = rdMolDescriptors.CalcRadiusOfGyration(mol)\n", " radii_gyration.append(rg)\n", " except:\n", " radii_gyration.append(np.nan)\n", " \n", " # Asphericity (0 = sphere, 1 = rod)\n", " try:\n", " asp = rdMolDescriptors.CalcAsphericity(mol)\n", " asphericity.append(asp)\n", " except:\n", " asphericity.append(np.nan)\n", " \n", " radii_gyration = np.array(radii_gyration)\n", " asphericity = np.array(asphericity)\n", " \n", " # Filter NaN\n", " valid = ~np.isnan(radii_gyration) & ~np.isnan(asphericity)\n", " \n", " if valid.sum() > 0:\n", " rg_avg = boltzmann_average_property(radii_gyration[valid], populations[valid])\n", " asp_avg = boltzmann_average_property(asphericity[valid], populations[valid])\n", " \n", " return {\n", " \"R_gyration_avg\": rg_avg,\n", " \"asphericity_avg\": asp_avg,\n", " \"R_gyration_range\": (radii_gyration[valid].min(), radii_gyration[valid].max()),\n", " \"asphericity_range\": (asphericity[valid].min(), asphericity[valid].max()),\n", " }\n", " \n", " return None\n", "\n", "\n", "if 'analysis' in dir():\n", " props = calculate_conformer_properties(\n", " analysis['mols'], \n", " analysis['populations']\n", " )\n", " \n", " if props:\n", " print(\"\\nBoltzmann-Averaged Properties:\")\n", " print(f\" Radius of gyration: {props['R_gyration_avg']:.2f} Å\")\n", " print(f\" Range: {props['R_gyration_range'][0]:.2f} - {props['R_gyration_range'][1]:.2f} Å\")\n", " print(f\" Asphericity: {props['asphericity_avg']:.3f}\")\n", " print(f\" Range: {props['asphericity_range'][0]:.3f} - {props['asphericity_range'][1]:.3f}\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 4. Temperature Effects on Populations\n", "\n", "Higher temperatures flatten the population distribution." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "def analyze_temperature_dependence(energies_rel, temperatures):\n", " \"\"\"\n", " Analyze how conformer populations change with temperature.\n", " \"\"\"\n", " results = []\n", " \n", " for T in temperatures:\n", " pops = calculate_boltzmann_populations(energies_rel, T)\n", " S_conf = calculate_conformational_entropy(pops)\n", " effective_n = np.exp(-np.sum(pops * np.log(pops + 1e-10)))\n", " \n", " # Population of global minimum\n", " p_global_min = pops[np.argmin(energies_rel)]\n", " \n", " results.append({\n", " \"T\": T,\n", " \"S_conf\": S_conf,\n", " \"effective_n\": effective_n,\n", " \"p_global_min\": p_global_min * 100\n", " })\n", " \n", " return pd.DataFrame(results)\n", "\n", "\n", "if 'analysis' in dir():\n", " temps = [200, 250, 298, 310, 350, 400, 500]\n", " df_temp = analyze_temperature_dependence(analysis['energies_rel'], temps)\n", " \n", " print(\"\\nTemperature Dependence of Conformer Distribution:\")\n", " print(\"-\" * 60)\n", " print(df_temp.to_string(index=False, float_format=\"%.1f\"))\n", " print(\"-\" * 60)\n", " print(\"\\nNote: At higher T, populations become more uniform.\")\n", " print(\"Body temperature (310K) vs room temperature (298K) matters!\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 5. Conformational Clustering\n", "\n", "Group similar conformers to identify distinct conformational families." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "def cluster_conformers_by_rmsd(mols, populations, rmsd_threshold=1.0):\n", " \"\"\"\n", " Cluster conformers by RMSD similarity.\n", " \n", " Returns list of clusters, each with representative and total population.\n", " \"\"\"\n", " from rdkit.Chem import AllChem\n", " \n", " n_mols = len(mols)\n", " \n", " # Simple greedy clustering\n", " clusters = [] # [(representative_idx, [member_indices])]\n", " assigned = set()\n", " \n", " # Sort by population (start with most populated)\n", " sorted_idx = np.argsort(populations)[::-1]\n", " \n", " for i in sorted_idx:\n", " if i in assigned:\n", " continue\n", " if mols[i] is None:\n", " continue\n", " \n", " # Start new cluster\n", " cluster_members = [i]\n", " assigned.add(i)\n", " \n", " # Find similar conformers\n", " for j in sorted_idx:\n", " if j in assigned or mols[j] is None:\n", " continue\n", " \n", " try:\n", " rmsd = AllChem.GetBestRMS(mols[i], mols[j])\n", " if rmsd < rmsd_threshold:\n", " cluster_members.append(j)\n", " assigned.add(j)\n", " except:\n", " continue\n", " \n", " clusters.append((i, cluster_members))\n", " \n", " # Calculate cluster populations\n", " cluster_data = []\n", " for rep_idx, members in clusters:\n", " total_pop = sum(populations[m] for m in members)\n", " cluster_data.append({\n", " \"representative\": rep_idx,\n", " \"n_members\": len(members),\n", " \"population\": total_pop * 100\n", " })\n", " \n", " return pd.DataFrame(cluster_data).sort_values(\"population\", ascending=False)\n", "\n", "\n", "if 'analysis' in dir() and len(analysis['mols']) > 1:\n", " try:\n", " df_clusters = cluster_conformers_by_rmsd(\n", " analysis['mols'], \n", " analysis['populations'],\n", " rmsd_threshold=1.0\n", " )\n", " \n", " print(\"\\nConformational Families (RMSD < 1.0 Å):\")\n", " print(\"-\" * 50)\n", " print(df_clusters.head(10).to_string(index=False))\n", " print(\"-\" * 50)\n", " print(f\"Total families: {len(df_clusters)}\")\n", " except Exception as e:\n", " print(f\"Clustering failed: {e}\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 6. Free Energy Surface\n", "\n", "The Boltzmann-weighted free energy is:\n", "\n", "$$G = -RT\\ln Z = -RT\\ln\\sum_i e^{-E_i/RT}$$\n", "\n", "This is the reference for calculating free energy differences." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "def calculate_free_energy_from_ensemble(energies_rel, T=298.15):\n", " \"\"\"\n", " Calculate the configurational free energy from conformer ensemble.\n", " \n", " G = -RT ln(Z) where Z = Σ exp(-E_i/RT)\n", " \"\"\"\n", " RT = R * T\n", " \n", " # Partition function\n", " Z = np.sum(np.exp(-energies_rel / RT))\n", " \n", " # Free energy (relative to global minimum)\n", " G = -RT * np.log(Z)\n", " \n", " # Entropy contribution\n", " populations = calculate_boltzmann_populations(energies_rel, T)\n", " S_conf = calculate_conformational_entropy(populations) / 1000 # kcal/(mol·K)\n", " \n", " # Average energy\n", " E_avg = np.sum(populations * energies_rel)\n", " \n", " return {\n", " \"G\": G,\n", " \"E_avg\": E_avg,\n", " \"S_conf\": S_conf * 1000, # cal/(mol·K)\n", " \"-T*S\": -T * S_conf,\n", " \"Z\": Z\n", " }\n", "\n", "\n", "if 'analysis' in dir():\n", " fe = calculate_free_energy_from_ensemble(analysis['energies_rel'])\n", " \n", " print(\"\\nFree Energy Analysis:\")\n", " print(\"=\" * 50)\n", " print(f\"Average energy : {fe['E_avg']:>8.2f} kcal/mol\")\n", " print(f\"Conformational entropy S: {fe['S_conf']:>8.2f} cal/(mol·K)\")\n", " print(f\"Entropy contribution -TS: {fe['-T*S']:>8.2f} kcal/mol\")\n", " print(f\"Free energy G (rel): {fe['G']:>8.2f} kcal/mol\")\n", " print(f\"Partition function Z: {fe['Z']:>8.2f}\")\n", " print(\"=\" * 50)\n", " print(\"\\nNote: G = - TS (approximately)\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 7. Comparison with Single-Conformer Analysis\n", "\n", "Demonstrates the importance of ensemble averaging." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "if 'analysis' in dir():\n", " # Compare single conformer vs ensemble\n", " props = calculate_conformer_properties(\n", " analysis['mols'], \n", " analysis['populations']\n", " )\n", " \n", " if props and len(analysis['mols']) > 0:\n", " # Single conformer (global minimum)\n", " idx_min = np.argmin(analysis['energies_rel'])\n", " mol_min = analysis['mols'][idx_min]\n", " \n", " if mol_min:\n", " try:\n", " rg_single = rdMolDescriptors.CalcRadiusOfGyration(mol_min)\n", " asp_single = rdMolDescriptors.CalcAsphericity(mol_min)\n", " \n", " print(\"\\nSingle Conformer vs Ensemble Comparison:\")\n", " print(\"-\" * 50)\n", " print(f\"{'Property':<25} {'Single':<12} {'Ensemble':<12}\")\n", " print(\"-\" * 50)\n", " print(f\"{'Radius of gyration (Å)':<25} {rg_single:<12.2f} {props['R_gyration_avg']:<12.2f}\")\n", " print(f\"{'Asphericity':<25} {asp_single:<12.3f} {props['asphericity_avg']:<12.3f}\")\n", " print(\"-\" * 50)\n", " print(\"\\nUsing only the global minimum can give misleading results!\")\n", " except Exception as e:\n", " print(f\"Property calculation failed: {e}\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 8. Practical Applications\n", "\n", "### NMR Analysis\n", "- J-couplings depend on dihedral angles\n", "- NOE intensities depend on interatomic distances\n", "- Observed values are Boltzmann averages\n", "\n", "### Drug Design\n", "- Binding affinities should include conformational penalty\n", "- Rigid molecules have entropic advantage\n", "- Bioactive conformation may not be global minimum\n", "\n", "### QSAR/ML\n", "- Use Boltzmann-averaged descriptors\n", "- Consider conformational entropy as descriptor\n", "- Multiple conformers may be needed for 3D-QSAR" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# Cleanup\n", "if 'drug_file' in dir() and os.path.exists(drug_file):\n", " os.unlink(drug_file)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Summary\n", "\n", "This tutorial covered:\n", "\n", "1. **Boltzmann distribution** - population = exp(-E/RT) / Z\n", "2. **Conformational entropy** - S = -R Σ p·ln(p)\n", "3. **Ensemble analysis** - effective number of conformers\n", "4. **Temperature effects** - higher T = flatter distribution\n", "5. **Boltzmann averaging** - = Σ p·property\n", "6. **Free energy** - G = -RT ln(Z)\n", "\n", "Key takeaways:\n", "- **1.36 kcal/mol ≈ 10:1 ratio** at room temperature\n", "- **Ensemble matters** - single conformer can be misleading\n", "- **Temperature matters** - body temp (310K) vs room temp (298K)" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3.10.0" } }, "nbformat": 4, "nbformat_minor": 4 }