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A comprehensive description of the protein should include its structure, thermodynamics, and kinetic properties. The recent rise of cryogenic electron microscopy (cryo-EM) provides new opportunities for the thermodynamic and kinetic research of proteins. There have been some researches in which cryo-EM is used not only to resolve the high-resolution structure of proteins but also to analyze the conformational distribution of proteins to infer their thermodynamic properties based on data processing methods. However, whether cryo-EM can be used to directly quantify the kinetics of proteins is still unclear. In this work, an ideal protein system, cyanobacterial circadian clock protein, is selected to explore the potential of cryo-EM used to analyze the non-equilibrium process of proteins. Previous research has illustrated that cryoelectron microscope can be used to infer the thermodynamic information about the KaiC protein such as the inter-subunit interaction within the hexamers. Herein, we extend the equilibrium Ising model of KaiC hexamers to a non-equilibrium statistical physics model, revealing the properties of the non-equilibrium process of KaiC hexamers. According to the non-equilibrium model and previous biochemical research, we find that the intrinsic properties of KaiC protein allow its non-equilibrium conformational distribution to be measured by cryo-EM.
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图 1 蓝藻生物钟蛋白 (a) 蓝藻生物钟核心蛋白的工作过程示意图[29,30]; (b) KaiC-CⅡ结构域的冷冻电镜结构的表面示意图, KaiC-AA六聚体中心孔径比KaiC-EE大[38]; (c) A-loop(氨基酸486-497)的两个构象, 暴露态(exposed state)和埋藏态(buried state)
Figure 1. Overview of cyanobacterial oscillator: (a) A carton illustration of cyanobacterial oscillator[29,30]; (b) space-filling depictions of structures of KaiC viewed from the CⅡ side, the central pore diameter of KaiC-AA is larger than that of KaiC-EE[38]; (c) two conformations of A-loop region, exposed state and buried state.
图 2 冷冻电镜解析的蓝藻生物钟蛋白KaiC的A-loop分布[38](a) KaiC-AA的A-loop的构象分布, 横坐标代表六聚体的13种构型, 空白圆圈表示暴露态, 蓝色填充圆圈表示埋藏态; (b) KaiC-EE的A-loop的构象分布
Figure 2. A-loop conformational distribution of KaiC hexamers[38]: (a) Conformational distribution of KaiC-AA hexamer, where 13 configurations are drawn within the x axis, un-filled circle represents exposed state and blue-filled circle represents buried state; (b) conformational distribution of KaiC-EE hexamer.
图 3 非平衡态Ising模型数值模拟结果 (a) J = 0.3时, 不同α条件下构象均值的演化(左), 均值m与外场B的关系(右); (b) J = 1.0时, 不同α条件下构象均值的演化(左), 均值m与外场B的关系(右)
Figure 3. Typical numerical simulation results of non-equilibrium Ising model: (a) evolution of m(t) when J = 0.3 in various α (left), relation between expectation value m and external filed B (right); (b) evolution of m(t) when J = 1.0 in various α (left), same as panel (a) (right).
图 4 (10)式对六聚体系统的适用性验证 (a) 外场振幅Bamp = 0.3时数值结果与(10)式的拟合, 图中实线为理论结果, 十字叉为数值模拟结果; (b) 不同参数范围时(10)式与数值模拟结果的拟合程度
Figure 4. Validation of applicability of Eq. (10) for the hexamer system: (a) Fitting the Eq. (10) with numerical results when the amplitude of external field Bamp is 0.3, here solid curve represents results of the theoretical equation whereas the cross represents numerical results; (b) fitting of the theoretical equation and numerical result in a wide range of parameters.
图 5 KaiC蛋白A-loop分布模拟结果 (a) KaiC蛋白A-loop构象分布模拟, 其中序号1—13与图2中六聚体构型依次对应, 红色虚线为p12的极大值对应的位置; (b) 非平衡态分布与平衡态分布的决定系数R2, 蓝色阴影部分为KaiC蛋白的参数范围
Figure 5. Simulation results of A-loop distribution for KaiC hexamers: (a) Evolution of A-loop distribution, where 13 states correspond to the markers drawn in Fig. 2 and red dash represents peak of distribution p12; (b) R2 between the non-equilibrium distribution and the corresponding equilibrium distribution, where blue shadow covers the parameter range of KaiC.
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[1] Henzler-Wildman K, Kern D 2007 Nature 450 964
Google Scholar
[2] Kern D 2021 Nat. Methods 18 435
Google Scholar
[3] Wei G, Xi W, Nussinov R, Ma B 2016 Chem. Rev. 116 6516
Google Scholar
[4] Motlagh H N, Wrabl J O, Li J, Hilser V J 2014 Nature 508 331
Google Scholar
[5] Dou Y, Dhatt-Gauthier K, Bishop K J M 2019 Curr. Opin. Solid State Mater. Sci. 23 28
Google Scholar
[6] Brown A I, Sivak D A 2020 Chem. Rev. 120 434
Google Scholar
[7] Fang X, Wang J 2020 Ann. Rev. Biophys. 49 227
Google Scholar
[8] Seifert U 2019 Annu. Rev. Condens. Matter Phys. 10 171
Google Scholar
[9] Fang X, Kruse K, Lu T, Wang J 2019 Rev. Mod. Phys. 91 045004
Google Scholar
[10] Ciliberto S 2017 Phys. Rev. X 7 021051
Google Scholar
[11] Bustamante C, Yan S 2022 Q. Rev. Biophys. 55 e9
Google Scholar
[12] Zhang D L, Ouyang Q 2021 Entropy 23 271
Google Scholar
[13] Frank J 2013 Biopolymers 99 832
Google Scholar
[14] Doerr A 2016 Nat. Methods 13 23
Google Scholar
[15] Sigworth F J 2007 Nat. Methods 4 20
Google Scholar
[16] Bonomi M, Vendruscolo M 2019 Curr. Opin. Struct. Biol. 56 37
Google Scholar
[17] Wu Z L, Chen E B, Zhang S W, Ma Y P, Mao Y D 2022 Int. J. Mol. Sci. 23 8872
Google Scholar
[18] Scheres S H 2016 Methods Enzymol. 579 125
Google Scholar
[19] Zhong E D, Bepler T, Berger B, Davis J H 2021 Nat. Methods 18 176
Google Scholar
[20] Huang R, Ripstein Z A, Rubinstein J L, Kay L E 2020 Angew. Chem. Int. Ed. Engl. 59 22423
Google Scholar
[21] Roh S H, Hryc C F, Jeong H H, Fei X, Jakana J, Lorimer G H, Chiu W 2017 Proc. Natl. Acad. Sci. U. S. A. 114 8259
Google Scholar
[22] Zhao Y Y, Schmid M F, Frydman J, Chiu W 2021 Nat. Commun. 12 4754
Google Scholar
[23] Amann S J, Keihsler D, Bodrug T, Brown N G, Haselbach D 2023 Structure 31 4
Google Scholar
[24] Bhattacharjee S, Feng X, Maji S, Dadhwal P, Zhang Z, Brown Z P, Frank J 2024 Cell 187 782
Google Scholar
[25] Zhang S W, Zou S T, Yin D Y, Zhao L H, Finley D, Wu Z L, Mao Y D 2022 Nature 605 567
Google Scholar
[26] Loveland A B, Demo G, Korostelev A A 2020 Nature 584 640
Google Scholar
[27] Nakajima M, Imai K, Ito H, Nishiwaki T, Murayama Y, Iwasaki H, Oyama T, Kondo T 2005 Science 308 414
Google Scholar
[28] Chavan A G, Swan J A, Heisler J, Sancar C, Ernst D C, Fang M, Palacios J G, Spangler R K, Bagshaw C R, Tripathi S, Crosby P, Golden S S, Partch C L, LiWang A 2021 Science 374 eabd4453
Google Scholar
[29] Partch C L 2020 J. Mol. Biol. 432 3426
Google Scholar
[30] Swan J A, Golden S S, LiWang A, Partch C L 2018 J. Biol. Chem. 293 5026
Google Scholar
[31] Rust M J, Markson J S, Lane W S, Fisher D S, O’Shea E K 2007 Science 318 809
Google Scholar
[32] Zhang D L, Cao Y S, Ouyang Q, Tu Y H 2020 Nat. Phys. 16 95
Google Scholar
[33] Paijmans J, Lubensky D K, Ten Wolde P R 2017 PLoS Comp. Biol. 13 e1005415
Google Scholar
[34] van Zon J S, Lubensky D K, Altena P R, ten Wolde P R 2007 Proc. Natl. Acad. Sci. U. S. A. 104 7420
Google Scholar
[35] Kim Y I, Dong G, Carruthers C W, Jr., Golden S S, LiWang A 2008 Proc. Natl. Acad. Sci. U.S.A. 105 12825
Google Scholar
[36] Tseng R, Chang Y G, Bravo I, Latham R, Chaudhary A, Kuo N W, Liwang A 2014 J. Mol. Biol. 426 389
Google Scholar
[37] Furuike Y, Mukaiyama A, Ouyang D, Ito-Miwa K, Simon D, Yamashita E, Kondo T, Akiyama S 2022 Sci Adv 8 eabm8990
Google Scholar
[38] Han X, Zhang D L, Hong L, Yu D Q, Wu Z L, Yang T, Rust M, Tu Y H, Ouyang Q 2023 Nat. Commun. 14 5907
Google Scholar
[39] Swan J A, Sandate C R, Chavan A G, Freeberg A M, Etwaru D, Ernst D C, Palacios J G, Golden S S, Liwang A, Lander G C, Partch C L 2022 Nat. Struct. Mol. Biol. 29 759
Google Scholar
[40] Han X, Wu Z L, Yang T, Ouyang Q 2022 Chin. Phys. Lett. 39 070501
Google Scholar
[41] Glauber R J 1963 J. Math. Phys. 4 294
Google Scholar
[42] Gillespie D T 2007 Annu. Rev. Phys. Chem. 58 35
Google Scholar
[43] Terauchi K, Kitayama Y, Nishiwaki T, Miwa K, Murayama Y, Oyama T, Kondo T 2007 Proc. Natl. Acad. Sci. U. S. A. 104 16377
Google Scholar
[44] Abe J, Hiyama T B, Mukaiyama A, Son S, Mori T, Saito S, Osako M, Wolanin J, Yamashita E, Kondo T, Akiyama S 2015 Science 349 312
Google Scholar
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