3.1. Matrix Multiplication

In this section, we will confirm that the data from SEC-SAXS experiments can be represented as matrices which are products of the curves. This fact is expressed in a formula as follows.

\[M = P \cdot C \qquad \qquad (1) \]

where

• \( M \) : measured data

• \( P \) : spectral curves

• \( C \) : elutional curves

• \( P \cdot C \) : matrix multiplication

Note

We refer to the curves related to elution as ‘elutional curves’ for contrast with ‘spectral curves.

3.1.1. Case of Single Component

For a case with a single component, confirm the following points.

• The \(P\) matrix consists of a single column vector which represents the spectral curve.

• The \(C\) matrix consists of a single row vector which represents the eluional curve.

import numpy as np
import matplotlib.pyplot as plt
from molass.SAXS.Models.Simple import guinier_porod
from molass.SEC.Models.Simple import gaussian

x = np.arange(300)
q = np.linspace(0.005, 0.5, 400)

def plot_single_component_data(scatter_params, elution_params):
    G, Rg, d = scatter_params
    h, m, s = elution_params
    fig = plt.figure(figsize=(15,4))
    ax1 = fig.add_subplot(131)
    ax2 = fig.add_subplot(132)
    ax3 = fig.add_subplot(133, projection='3d')
    
    I, q1 = guinier_porod(q, G, Rg, d, return_also_q1=True)
    # ax1.set_yscale('log')
    ax1.set_title("Scattering Curve: P")
    ax1.plot(q, I)
    ax1.axvline(q1, linestyle=':', color="green", label='Guinier-Porod Boundary')
    ax1.legend()

    y = gaussian(x, h, m, s)
    ax2.set_title("Elution Curve: C")
    ax2.plot(x, y)
    
    P = I.reshape((len(q),1))   # make it a matrix
    C = y.reshape((1,len(x)))   # make it a matrix
    M = P @ C                   # matrix multiplication
    xx, qq = np.meshgrid(x, q)
    ax3.set_title("3D Data View: M=PC")
    ax3.plot_surface(qq, xx, M)
    fig.tight_layout()

plot_single_component_data((1, 35, 3), (1, 150, 30))

3.1.2. Case of Multiple Components

For a case with multiple components, confirm the following points.

• The \(P\) matrix consists of multiple column vectors which represent the spectral curves.

• The \(C\) matrix consists of multiple row vectors which represent the eluional curves.

def plot_multiple_component_data(scattering_params, elution_params, use_matrices=False, view=None):
    fig = plt.figure(figsize=(15,5))
    ax1 = fig.add_subplot(131)
    ax2 = fig.add_subplot(132)
    ax3 = fig.add_subplot(133, projection='3d')
    
    fig.suptitle(r"Illustration of Decomposition $M = P \cdot C$ simulated with Guinier-Prod and EGH Models in Molass Library", fontsize=16)

    # ax1.set_yscale('log')
    ax1.set_title("Scattering Curves in P", fontsize=14)
    ax1.set_xlabel("Q")
    ax1.set_ylabel("Intensity")
    w_list = []
    rgs = []
    for i, (G, Rg, d) in enumerate(scattering_params):
        I, q1 = guinier_porod(q, G, Rg, d, return_also_q1=True)
        w_list.append(I)
        color = "C%d" % i
        ax1.plot(q, I, color=color, label='component-%d: $R_g=%.3g$' % (i+1, Rg))
        rgs.append(Rg)
        # ax1.axvline(q1, linestyle=':', color=color, label='Guinier-Porod $Q_1$')
    ax1.legend()

    y_list = []
    ax2.set_title("Elution Curves in C", fontsize=14)
    ax2.set_xlabel("Frames")
    ax2.set_ylabel("Concentration")
    for i, (h, m, s) in enumerate(elution_params):
        y = gaussian(x, h, m, s)
        y_list.append(y)
        ax2.plot(x, y, label='component-%d: $R_g=%.3g$' % (i+1, rgs[i]))
    ty = np.sum(y_list, axis=0)
    ax2.plot(x, ty, ':', color='red', label='total')
    ax2.legend()

    ax3.set_title(r"$M$ calculated by $P \cdot C$", fontsize=14)
    ax3.set_xlabel("Q")
    ax3.set_ylabel("Frames")
    ax3.set_zlabel("Intensity")
    xx, qq = np.meshgrid(x, q)
    if use_matrices:
        P = np.array(w_list).T
        C = np.array(y_list)
        M = P @ C
    else:
        zz_list = []
        for w, y in zip(w_list, y_list):
            zz =  w.reshape((len(q),1)) @ y.reshape((1,len(x)))
            zz_list.append(zz)
        M = np.sum(zz_list, axis=0)
    ax3.plot_surface(qq, xx, M, color='red', alpha=0.5)
    # ax3.legend()
    if view is not None:
        ax3.view_init(azim=view[0], elev=view[1])    
    fig.tight_layout()
    fig.subplots_adjust(right=0.95)

rgs = (35, 32, 23)
scattering_params = [(1, rgs[0], 3), (1, rgs[1], 3), (1, rgs[2], 4)]
elution_params = [(1, 100, 12), (0.3, 125, 16), (0.5, 200, 16)]
plot_multiple_component_data(scattering_params, elution_params, use_matrices=True, view=(-15, 15))