{"id":332,"date":"2020-03-19T15:40:29","date_gmt":"2020-03-19T15:40:29","guid":{"rendered":"http:\/\/statisticalbiophysicsblog.org\/?p=332"},"modified":"2020-03-25T18:36:25","modified_gmt":"2020-03-25T18:36:25","slug":"squeezing-physics-from-math-exercise-2","status":"publish","type":"post","link":"https:\/\/statisticalbiophysicsblog.org\/?p=332","title":{"rendered":"Squeezing physics from math (Exercise 2)"},"content":{"rendered":"

To become stronger in theory, we do math. But we need to understand mathematical results in physical terms. The math and the conceptual picture must reinforce one another in our minds, or we\u2019ll forget both. We also have to understand the assumptions underlying solvable models.<\/p>\n

So it\u2019s time for the solution to the previous problem \u2026 and new questions to understand that solution.<\/p>\n

<\/p>\n

Last post<\/a>, you were asked to solve the two-state system,<\/p>\n

<\/a><\/p>\n

(1) <\/span>   <\/span>\"\begin{equation*}<\/p>\n

where \"k_{AB}\" is the reaction rate constant for the transition from A to B and likewise for \"k_{BA}\". States A and B represent two configurational states of a system like a protein, cell, solid material, or maybe a weather or epidemiological system. The time-dependent populations of states A and B, \"P_A(t)\" and \"P_B(t)\" are governed by<\/p>\n

<\/a><\/p>\n

(2) <\/span>   <\/span>\"\begin{align*}<\/p>\n

Hopefully, you have written down the solutions, which are<\/p>\n

<\/a><\/p>\n

(3) <\/span>   <\/span>\"\begin{align*}<\/p>\n

where \"k = k_{AB} + k_{BA}\". You should be able to figure out the equilibrium fractions, \"P_A^\mathrm{eq}\" and \"P_B^\mathrm{eq}\". As needed, a full discussion of this system is available<\/a>.<\/p>\n

Now that we know the math, the physics questions begin:<\/p>\n

    \n
  1. \n Why is the relaxation time \"1/k\" governed by the rates in both directions?<\/li>\n
  2. \n What is the physical meaning of the reciprocals of the rates?<\/li>\n
  3. \n In qualitative terms, how can the solutions to the conformational A-B problem above help us understand binding kinetics? Hint: Consider the differences between binding and conformational ODEs.<\/li>\n
  4. \n What assumptions are built into the two-state model? Equivalently, in what ways can a two-state model fail<\/em> to describe the behavior of a complex system?<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"

    To become stronger in theory, we do math. But we need to understand mathematical results in physical terms. The math and the conceptual picture must reinforce one another in our minds, or we\u2019ll forget both. We also have to understand the assumptions underlying solvable models. So it\u2019s time for the solution to the previous problem […]<\/p>\n","protected":false},"author":6,"featured_media":345,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[22,11,19],"tags":[],"class_list":["post-332","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-exercises","category-general-biophysics","category-non-equilibrium-physics"],"_links":{"self":[{"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=\/wp\/v2\/posts\/332","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=\/wp\/v2\/users\/6"}],"replies":[{"embeddable":true,"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=332"}],"version-history":[{"count":5,"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=\/wp\/v2\/posts\/332\/revisions"}],"predecessor-version":[{"id":337,"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=\/wp\/v2\/posts\/332\/revisions\/337"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=\/wp\/v2\/media\/345"}],"wp:attachment":[{"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=332"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=332"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=332"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}