A Study of Polymorphic Sickle Hemoglobin Polymers
James Amarando, Frank A. Ferrone
Department of Physics, Drexel University
Introduction
Hemoglobin (Hb) is a protein found in human red blood
cells (RBCs). Hb is responsible for the transport of oxygen
within the blood. Sickle Hemoglobin (HbS) is a mutation of
Hb in which a Glu is replaced by a Val on the two beta
subunits of the molecule. This mutation causes HbS to
polymerize inside the RBCs. These polymers are long and
rigid and cause the RBCs to become sickle shaped. It also
makes them too rigid to deform to pass through thin
capillaries, causing many complications such as strokes and
anemia.
An occluded red blood cell in the capillary
blocks blood flow
Axial contacts occur at locations between molecules as
they stack on each other going up the polymer. The
interaction between the axials is not as strong as the
laterals. The axial contacts occur on the sub protein chains
of the molecule.
Method – 3D Images Using Visual Python
The Flex Angle
Given the (x,y,z) coordinates for the contact points on one
hemoglobin, code was written which will generate a full
polymer. Figure 5 has an example of one molecule using the
bead molecule. In this the red and blue spheres are lateral
contacts (both the valine and the “pocket”). The other colored
beads are the axial contacts, with colors matching those
identities in figure 4. The white sphere on each pole represents
a repulsion, meaning no two white spheres should ever be
touching.
The flex angle mathematically is a rotation at an arbitrary point
around an arbitrary axis. This is coded in Cartesian
coordinates. The conversion is accomplished using the
Quaternions to derive a rotation matrix about an arbitrary
normalized axis a= (ax,ay,az,).
θ
Figure 8: Quaternion derived flex angle
Note: s and c are shorthand for sin(θ) and cos(θ)
Figure 3: Model of the contact points
Results
Bogdan Barz, Brigita Urbanc and Frank A. Ferrone,
(unpublished) proposed that the same amino acids can be used
in a different pairing (figure 4).
Figure 6: One hemoglobin bead
(output from code)
Oxygen will not be delivered to tissues beyond
the blockage (blue)
Figure 1: Sickle cell disease illustration
Hb is composed of two alpha chains and two beta chains.
Source: http://www.bio.miami.edu/~cmallery/150/chemistry/hemoglobin.jpg
Working with just a single pair, a polymer of 14 beads is
generated using the new axial points. This is figure 9.
Algorithm
Figure 4: Left – old axial contacts
Right – new axial contacts
Given an alternative set of axial contact points, can a
feasible model of an HbS polymer be generated?
To make a polymer, a second bead is connected to the first
using lateral contacts. A flex angle is then put in about the
lateral contact so that a third bead's axials will align with the
desired axials on the first beads.
From a coding perspective, the math of adding in new beads
is performed in Matlab, which then outputs coordinates in
visual python formatting to the end of a python script. The
bash wrapper script will execute the python script once all
beads are generated. A high level flow chart of the algorithm
is in figure 7.
Figure 9: HbS polymer using new contacts
(output from code)
This result is a linear polymer (linear in the sense of it is
essentially planar). The next steps is to correlate this result to
the expected result of a twist. Figure 10 is previous work
assembled using the same contact points but using styrofoam
balls. It is expected that this twist can be achieved by altering
the bend of the flex angle at the lateral contacts.
The bead model is used in simulations of the polymer. In a bead
model, each hemoglobin is represented by a sphere.
Figure 2: Structure of Hb
The HbS polymer is formed as the hydrophobic valine on
one molecule comes into contact with another hydrophobic
“pocket” on another molecule. This forms a very strong
hydrophobic bond which is similar to a ball and socket joint.
These are referred to as the lateral contacts, as they connect
molecules within a cross section. These contacts are well
defined and are the strongest intermolecular interactions
within the polymer.
The ultimate goal is shown in figure 5 , which is based on
electron microscopy , which cannot distinguish the contact pairs.
Figure 10: Styrofoam model of HbS polymer
(Klaiss and Ferrone, unpublished)
Figure 5: Bead model polymer
Figure 7: Algorithm Flow Chart