a dimensionless temperature, T, defined as the
thermal energy of the system divided by the bonding energy of a particular interaction. A previous
study measured the Gibbs free energy of hydrogen
bonding between hydrated strandlike peptides to
average 2.6 kcal/H-bond/mol (27). For links consisting of 3 to 8 H-bonds at 10°C, T = 0.08 to 0.03. This
estimate of T applies to the 100 and 80% radius
region of the lens, where linkages between proteins
are primarily between disordered loops.
At radial positions less than 60%, proteins form
ramifying lamellae and then a proteinaceous material with no bulk water. These structures are
consistent with patchy systems in which patches
have become so large and/or numerous that their
interactions approach those of isotropic, spherical
particles (Figs. 3D and 4D). Here, the energies of
protein-protein binding are likely similar to those
involved in protein docking, in the range of 10 to
20 kcal/bond/mol, so the lower bound for T for
these interactions is ~0.01 (28).
Our data show that at all radial positions and
every possible density, structure in the intact squid
lens are dominated by attractive interactions between S-crystallin dimers, a result that is inconsistent with isotropic potentials, because these
systems will undergo liquid-liquid phase separation and opacification at some point (29).
From a materials perspective, this lens could
in principle be built from a single mixture of particles with M slightly greater than two, since any
system to the right of the spinodal line for <M> =
2.1 in the patchy colloid phase diagram may form
a volumetric material with low density variation.
However, our data show that <M> changes monotonically with lens radius, from a minimum near
two in the periphery to a maximum near six at
the core of the lens.
There is a biological rationale for the lens evolving an array of <M> rather than simply using one
mixture with <M> ≈ 2.1 at many different densities. New lens cells must be filled gradually with
protein through the process of mRNAs and amino
acids diffusing to large ribosomes, with full-length
proteins then diffusing away from the ribosome.
In any cell whose proteome-determined <M> allowed for gelation at a density lower than the
appropriate endpoint for a given radial position,
protein synthesis would be arrested prematurely.
Patchy colloidal physics allows for a cellular mechanism by which lens cells gel near the optically
appropriate endpoint density for a given radial
position. The cellular transcriptomes systematically
titrate S-crystallin isoforms as <M> changes as a
function of lens radius. We hypothesize that a cell
then passively ceases protein synthesis at the appropriate density, because upon gelation, protein
synthesis would simply stop due to inability of
large polymers (mRNAs and newly transcribed
protein) to diffuse through the material.
Given that, in principle, one protein sequence
could encode one valence in the lens, requiring
a minimum of six S-crystallin sequences, there
were a large number of unique S-crystallin se-
quences encoded by the genome, and the loops
encoded by these sequences that form the link-
ages between proteins also had high chemical
polydispersity. We speculate that this high degree
of polydispersity in loop sequence and subsequent
linker interaction dynamics may both avoid kinetic
traps during nucleation and reduce the entropic
cost of forming a volume-spanning network (30).
This set of linker sequences generated by evolu-
tion may then help inform experimental attempts
to exploit the self-assembly principles revealed by
patchy particle theories.
We observed that the difference between the
original density of the lens and the density of the
pellet formed from low effective temperature bonds
increased from near zero at the periphery to a
factor of two at the core (Fig. 4, A to C). At the core
of the lens, the pellet after a dilution-induced
phase/state transition had a packing fraction near
0.5, indicating <M> for the system near six, or
square-lattice-like packing, whereas the intact lens
core has little or no bulk water and a packing
fraction approaching 1.0 (consistent with M = 12).
We speculate that in the squid embryo, these orig-
inal, most-central cells in the lens structure initially
gel near the lower square-lattice packing fraction
of 0.5, consistent with constituent proteins of <M> ≈
6. Then, as new cells are added to the organ periphery during lens growth, the newer material will
be both more compliant due to a lower coordination number and have a higher charge density
due to the increasingly positive surface charge of
the proteins in the peripheral layers (8). Both of
these factors will tend to cause water to migrate
outward from the lens core as the structure grows,
systematically increasing the density of central
regions of the lens with the animal’s age and size
and potentially resulting in a near-dry lens core
in the mature organ.
Our data also describe a gradual “inside-out”
mechanism for evolving a gradient index lens
from a single protein fold. In this view, an initial
protein closely related to enzymatic GST duplicated
for lens expression and formed a high-density
colloidal gel through near-isotropic interactions.
Selection pressure for a gradient index then resulted in proteins with less-isotropic interactions
and the resulting lamellar structures observed
in extant lenses, consistent with patchy colloidal systems of intermediate average valence (31).
Further selection on S-crystallins resulted in an
unstructured loop encoded in a novel exon that was
able to undergo flexible, nonspecific hydrogen-bonding interactions with other similar loops.
These loops, coupled to the increase in positive
charge on the folded, spherical surfaces of the protein, represent an evolutionary innovation of a
protein particle able to enforce a coordinate number of M = 2. With this loop-binding innovation,
squid lenses were then able to exploit the entire
physics of the patchy colloid phase diagram, and
the thermodynamically stable materials with low
density fluctuation that result from this physics.
It is also possible that this patchy colloidal per-
spective could provide insights into still poorly
understood aspects of vertebrate lens biology. In
particular, the polydisperse nature of alpha crys-
tallins in the low-density regions of the vertebrate
lens have been a puzzle; even in solution, they ap-
pear to interact via “tentacles” in ways that are hard
to specifically characterize (32). It is possible that
they are also acting as low-valence patchy colloidal
gels in vivo, as is the case for squid S-crystallins.
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We are grateful to R. Kamien and E. Eiser for useful discussions and
to D. Fox for assistance with data visualization. S. Johnsen’s comments
improved the manuscript. We are also grateful to an anonymous
reviewer whose thoughtful questions improved the work. Financial
support was provided by the National Science Foundation Materials
Research Science and Engineering Center DMR11-20901 to P.A.H.;
by the Packard Foundation Fellowship for Science and Engineering,
Sloan Foundation, NSF-1351935, Kaufman Foundation, and University
of Pennsylvania to A. M.S.; and by the Department of Defense’s National
Defense Science and Engineering Graduate fellowship program to
T.C.D. SAXS data are archived at Small Angle Scattering Biological Data
Bank under accession numbers SASDCQ5 to U5, and RNA-seq data
are archived at GenBank under SRR5528268–9.
Materials and Methods
Figs. S1 to S4
23 October 2016; resubmitted 6 February 2017
Accepted 7 July 2017