{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T20:00:20Z","timestamp":1790712020429,"version":"4.1.0"},"reference-count":112,"publisher":"American Physical Society (APS)","issue":"1","license":[{"start":{"date-parts":[[2025,3,27]],"date-time":"2025-03-27T00:00:00Z","timestamp":1743033600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000002","name":"National Institutes of Health","doi-asserted-by":"publisher","award":["T32GM145452"],"award-info":[{"award-number":["T32GM145452"]}],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000002","name":"National Institutes of Health","doi-asserted-by":"publisher","award":["T15LM007056-37"],"award-info":[{"award-number":["T15LM007056-37"]}],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"short-container-title":["PRX Life"],"accepted":{"date-parts":[[2025,3,3]]},"abstract":"<jats:p>\n                    Proteins fold to a specific functional conformation with a densely packed core that controls their stability. Despite their importance, we lack a quantitative explanation for why all protein cores, regardless of their overall fold, possess the same average packing fraction\n                    <a:math xmlns:a=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <a:mrow>\n                        <a:mo>\u2329<\/a:mo>\n                        <a:mi>\u03d5<\/a:mi>\n                        <a:mo>\u232a<\/a:mo>\n                        <a:mo>\u2248<\/a:mo>\n                        <a:mn>0.55<\/a:mn>\n                      <\/a:mrow>\n                    <\/a:math>\n                    . However, important developments in the physics of jamming in particulate systems can shed light on the packing of protein cores. Here, we extend the framework of jamming to describe core packing in collapsed polymers, as well as in all-atom models of folded proteins. First, we show in a spherical bead-spring polymer model (with and without bond-angle constraints) that as the hydrophobic interactions increase relative to thermal fluctuations, a jamming-like transition occurs when the core packing fraction exceeds\n                    <b:math xmlns:b=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <b:msub>\n                        <b:mi>\u03d5<\/b:mi>\n                        <b:mi>c<\/b:mi>\n                      <\/b:msub>\n                    <\/b:math>\n                    with the same power-law scaling behavior for the potential energy\n                    <c:math xmlns:c=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <c:msub>\n                        <c:mi>V<\/c:mi>\n                        <c:mi>r<\/c:mi>\n                      <\/c:msub>\n                    <\/c:math>\n                    , excess contact number\n                    <d:math xmlns:d=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <d:mrow>\n                        <d:mi mathvariant=\"normal\">\u0394<\/d:mi>\n                        <d:mi>N<\/d:mi>\n                      <\/d:mrow>\n                    <\/d:math>\n                    , and characteristic frequency of the vibrational density of states\n                    <f:math xmlns:f=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <f:msup>\n                        <f:mi>\u03c9<\/f:mi>\n                        <f:mo>*<\/f:mo>\n                      <\/f:msup>\n                    <\/f:math>\n                    versus\n                    <g:math xmlns:g=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <g:mrow>\n                        <g:mi mathvariant=\"normal\">\u0394<\/g:mi>\n                        <g:mi>\u03d5<\/g:mi>\n                        <g:mo>=<\/g:mo>\n                        <g:mi>\u03d5<\/g:mi>\n                        <g:mo>\u2212<\/g:mo>\n                        <g:msub>\n                          <g:mi>\u03d5<\/g:mi>\n                          <g:mi>c<\/g:mi>\n                        <\/g:msub>\n                      <\/g:mrow>\n                    <\/g:math>\n                    as that for jammed particulate systems. Then, we develop an all-atom model for proteins and find that, above\n                    <i:math xmlns:i=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <i:mrow>\n                        <i:msub>\n                          <i:mi>\u03d5<\/i:mi>\n                          <i:mi>c<\/i:mi>\n                        <\/i:msub>\n                        <i:mo>\u223c<\/i:mo>\n                        <i:mn>0.55<\/i:mn>\n                      <\/i:mrow>\n                    <\/i:math>\n                    , protein cores undergo a jamming-like transition, but with anomalous power-law scaling for\n                    <j:math xmlns:j=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <j:msub>\n                        <j:mi>V<\/j:mi>\n                        <j:mi>r<\/j:mi>\n                      <\/j:msub>\n                      <j:mo>,<\/j:mo>\n                      <j:mo>\u00a0<\/j:mo>\n                      <j:mrow>\n                        <j:mi mathvariant=\"normal\">\u0394<\/j:mi>\n                        <j:mi>N<\/j:mi>\n                      <\/j:mrow>\n                    <\/j:math>\n                    , and\n                    <l:math xmlns:l=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <l:msup>\n                        <l:mi>\u03c9<\/l:mi>\n                        <l:mo>*<\/l:mo>\n                      <\/l:msup>\n                    <\/l:math>\n                    versus\n                    <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <m:mrow>\n                        <m:mi mathvariant=\"normal\">\u0394<\/m:mi>\n                        <m:mi>\u03d5<\/m:mi>\n                      <\/m:mrow>\n                    <\/m:math>\n                    . The all-atom protein model remains close to the native protein structure during jamming and accurately refolds from partially unfolded states.\n                  <\/jats:p>","DOI":"10.1103\/prxlife.3.013018","type":"journal-article","created":{"date-parts":[[2025,3,27]],"date-time":"2025-03-27T15:35:04Z","timestamp":1743089704000},"update-policy":"https:\/\/doi.org\/10.1103\/crossmark-policy","source":"Crossref","is-referenced-by-count":4,"title":["Protein Folding as a Jamming Transition"],"prefix":"10.1103","volume":"3","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1588-2996","authenticated-orcid":true,"given":"Alex T.","family":"Grigas","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale University"},{"id":[{"id":"https:\/\/ror.org\/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale 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