Dihedral/Torsion Angle From Four Points in Cartesian Coordinates in Python

Calculate Dihedral/Torsion Angle from Four Cartesian Points in Python

Learn how to accurately compute the dihedral (torsion) angle between four points in 3D Cartesian coordinates using Python and NumPy, a fundamental operation in molecular modeling and structural analysis.

The dihedral angle, also known as a torsion angle, is a fundamental geometric property in chemistry, physics, and molecular biology. It describes the angle between two intersecting planes, specifically the angle between the plane defined by points P1, P2, P3 and the plane defined by points P2, P3, P4. This measurement is crucial for understanding molecular conformations, protein folding, and material properties. This article will guide you through calculating this angle efficiently using Python and the NumPy library.

Understanding the Dihedral Angle

Imagine four atoms (or points) in a chain: P1, P2, P3, and P4. The dihedral angle is the angle of rotation around the bond P2-P3. It's defined by the relative orientation of the plane containing P1, P2, P3 and the plane containing P2, P3, P4. A positive angle typically indicates a clockwise rotation when looking down the P2-P3 bond from P2 to P3, while a negative angle indicates a counter-clockwise rotation. The angle ranges from -180° to +180°.

graph TD
    subgraph Plane 1
        P1_1(P1) --> P2_1(P2)
        P2_1 --> P3_1(P3)
        P3_1 --> P1_1
    end
    subgraph Plane 2
        P2_2(P2) --> P3_2(P3)
        P3_2 --> P4_2(P4)
        P4_2 --> P2_2
    end
    P2_1 --- P2_2
    P3_1 --- P3_2
    style Plane 1 fill:#f9f,stroke:#333,stroke-width:2px
    style Plane 2 fill:#ccf,stroke:#333,stroke-width:2px
    linkStyle 0,2,3,5 stroke-width:0px
    linkStyle 1,4 stroke:#000,stroke-width:2px,fill:none
    classDef planeStyle fill:#f9f,stroke:#333,stroke-width:2px
    class Plane 1 planeStyle
    class Plane 2 planeStyle
    P1_1["P1 (x1,y1,z1)"]
    P2_1["P2 (x2,y2,z2)"]
    P3_1["P3 (x3,y3,z3)"]
    P4_2["P4 (x4,y4,z4)"]
    P2_2["P2 (x2,y2,z2)"]
    P3_2["P3 (x3,y3,z3)"]

Conceptual diagram of two planes defining a dihedral angle.

Mathematical Formulation

To calculate the dihedral angle, we first need to define three vectors:

1. b1 = P2 - P1
2. b2 = P3 - P2
3. b3 = P4 - P3

Next, we calculate the normal vectors to the two planes:

• n1 = b1 x b2 (normal to the P1-P2-P3 plane)
• n2 = b2 x b3 (normal to the P2-P3-P4 plane)

The dihedral angle θ can then be found using the dot product and cross product of these normal vectors, along with the central bond vector b2:

cos(θ) = (n1 ⋅ n2) / (|n1| |n2|)

However, this formula only gives the magnitude of the angle (0° to 180°). To get the signed angle (-180° to +180°), we use the atan2 function, which requires both the dot product and the scalar triple product:

θ = atan2((n1 x n2) ⋅ b2, n1 ⋅ n2)

Where (n1 x n2) ⋅ b2 is the scalar triple product, which determines the sign of the angle.

Python Implementation with NumPy

NumPy provides efficient array operations, making it ideal for vector calculations. We'll define a function that takes four 3D points as NumPy arrays and returns the dihedral angle in degrees.

import numpy as np

def calculate_dihedral_angle(p1, p2, p3, p4):
    """
    Calculates the dihedral angle (in degrees) between four points.

    Args:
        p1, p2, p3, p4 (np.array): 3D coordinates of the four points.

    Returns:
        float: The dihedral angle in degrees, ranging from -180 to 180.
    """
    # Convert points to NumPy arrays if they aren't already
    p1 = np.array(p1)
    p2 = np.array(p2)
    p3 = np.array(p3)
    p4 = np.array(p4)

    # Define the three vectors
    b1 = p2 - p1
    b2 = p3 - p2
    b3 = p4 - p3

    # Calculate normal vectors to the planes
    n1 = np.cross(b1, b2)
    n2 = np.cross(b2, b3)

    # Normalize normal vectors
    n1_unit = n1 / np.linalg.norm(n1)
    n2_unit = n2 / np.linalg.norm(n2)

    # Calculate the dot product of normalized normal vectors
    cos_theta = np.dot(n1_unit, n2_unit)

    # Ensure cos_theta is within valid range for arccos due to potential floating point errors
    cos_theta = np.clip(cos_theta, -1.0, 1.0)

    # Calculate the angle magnitude
    angle_rad = np.arccos(cos_theta)

    # Determine the sign of the angle using the scalar triple product
    # (n1 x n2) . b2
    sign = np.sign(np.dot(np.cross(n1, n2), b2))

    # Apply the sign to the angle
    dihedral_angle_rad = angle_rad * sign

    # Convert to degrees
    dihedral_angle_deg = np.degrees(dihedral_angle_rad)

    return dihedral_angle_deg

# Example Usage:
# Define four points (e.g., from a molecule)
point_A = [0.0, 0.0, 0.0]
point_B = [1.0, 0.0, 0.0]
point_C = [1.0, 1.0, 0.0]
point_D = [2.0, 1.0, 0.0]

angle1 = calculate_dihedral_angle(point_A, point_B, point_C, point_D)
print(f"Dihedral angle (planar, 0 degrees): {angle1:.2f} degrees")

# Example with a non-zero angle (e.g., staggered conformation)
point_E = [0.0, 0.0, 0.0]
point_F = [1.0, 0.0, 0.0]
point_G = [1.0, 1.0, 0.0]
point_H = [0.5, 1.0, 0.866] # Rotated around FG bond

angle2 = calculate_dihedral_angle(point_E, point_F, point_G, point_H)
print(f"Dihedral angle (staggered, approx 60 degrees): {angle2:.2f} degrees")

# Example with a negative angle
point_I = [0.0, 0.0, 0.0]
point_J = [1.0, 0.0, 0.0]
point_K = [1.0, 1.0, 0.0]
point_L = [0.5, 1.0, -0.866] # Rotated in the opposite direction

angle3 = calculate_dihedral_angle(point_I, point_J, point_K, point_L)
print(f"Dihedral angle (negative, approx -60 degrees): {angle3:.2f} degrees")

Python function to calculate the dihedral angle using NumPy.

Alternative using np.arctan2 for Signed Angle

A more direct way to get the signed angle is by using np.arctan2. This function takes two arguments: y and x, and returns the angle atan2(y, x) in radians. For the dihedral angle, y corresponds to the scalar triple product and x corresponds to the dot product of the normal vectors.

import numpy as np

def calculate_dihedral_angle_atan2(p1, p2, p3, p4):
    """
    Calculates the dihedral angle (in degrees) between four points using np.arctan2.

    Args:
        p1, p2, p3, p4 (np.array): 3D coordinates of the four points.

    Returns:
        float: The dihedral angle in degrees, ranging from -180 to 180.
    """
    p1 = np.array(p1)
    p2 = np.array(p2)
    p3 = np.array(p3)
    p4 = np.array(p4)

    b1 = p2 - p1
    b2 = p3 - p2
    b3 = p4 - p3

    n1 = np.cross(b1, b2)
    n2 = np.cross(b2, b3)

    # Calculate y (scalar triple product) and x (dot product)
    y = np.dot(np.cross(n1, b2), n2) # (n1 x b2) . n2 is equivalent to (n1 x n2) . b2
    x = np.dot(n1, n2)

    angle_rad = np.arctan2(y, x)
    dihedral_angle_deg = np.degrees(angle_rad)

    return dihedral_angle_deg

# Example Usage with atan2 version:
point_A = [0.0, 0.0, 0.0]
point_B = [1.0, 0.0, 0.0]
point_C = [1.0, 1.0, 0.0]
point_D = [2.0, 1.0, 0.0]

angle1_atan2 = calculate_dihedral_angle_atan2(point_A, point_B, point_C, point_D)
print(f"Dihedral angle (atan2, planar, 0 degrees): {angle1_atan2:.2f} degrees")

point_E = [0.0, 0.0, 0.0]
point_F = [1.0, 0.0, 0.0]
point_G = [1.0, 1.0, 0.0]
point_H = [0.5, 1.0, 0.866]

angle2_atan2 = calculate_dihedral_angle_atan2(point_E, point_F, point_G, point_H)
print(f"Dihedral angle (atan2, staggered, approx 60 degrees): {angle2_atan2:.2f} degrees")

point_I = [0.0, 0.0, 0.0]
point_J = [1.0, 0.0, 0.0]
point_K = [1.0, 1.0, 0.0]
point_L = [0.5, 1.0, -0.866]

angle3_atan2 = calculate_dihedral_angle_atan2(point_I, point_J, point_K, point_L)
print(f"Dihedral angle (atan2, negative, approx -60 degrees): {angle3_atan2:.2f} degrees")

Python function using np.arctan2 for a more concise signed dihedral angle calculation.

Considerations and Edge Cases

While the provided functions are robust, it's important to consider potential edge cases:

• Collinear Points: If P1, P2, P3 are collinear, or P2, P3, P4 are collinear, the cross product will result in a zero vector, meaning the normal vector is undefined. This typically happens when the bond vectors b1 or b3 are parallel to b2. The np.linalg.norm function will return 0, leading to division by zero errors. In such cases, the dihedral angle is undefined or ambiguous.
• Zero Central Bond: If P2 and P3 are the same point, the central bond b2 is a zero vector. This also makes the normal vectors undefined and the dihedral angle meaningless.

Robust implementations in production systems might include checks for these conditions to raise appropriate errors or return NaN.