Jamming proteins with slipknots and their free energy landscape Joanna I. SuGLYPH<suppress>lkowska $^{1}$, Piotr SuGLYPH<suppress>lkowski 2 , 3 , 4 and Jos´ e N. Onuchic 1 1 Center for Theoretical Biological Physics, University of California San Diego, Gilman Drive 9500, La Jolla 92037, 3 California Institute of Technology, Pasadena, CA 92215, 4 Institute for Nuclear Studies, Ho˙za 69, 00-681 Warsaw, Poland 2 Physikalisches Institute and Bethe Center for Theoretical Physics, Universit¨ at Bonn, Nussallee 12, 53115 Bonn, Germany Theoretical studies of stretching proteins with slipknots reveal a surprising growth of their un- folding times when the stretching force crosses an intermediate threshold. This behavior arises as a consequence of the existence of alternative unfolding routes that are dominant at different force ranges. Responsible for longer unfolding times at higher forces is the existence of an intermediate, metastable configuration where the slipknot is jammed. Simulations are performed with a coarsed grained model with further quantification using a refined description of the geometry of the slip- knots. The simulation data is used to determine the free energy landscape (FEL) of the protein, which supports recent analytical predictions. PACS numbers: 87.15.ap, 87.14.E-, 87.15.La, 82.37.Gk, 87.10.+e dine kinase (PDB code: 1e2i [17]). The large increase in determining new protein struc- tures has led to the discovery of several proteins with complicated topology. This new fact has arised the ques- tion if their energy landscape and the folding mechanism is similar to typical proteins. One class of such proteins includes knotted proteins which comprise around 1% of all structures deposited in the PDB database [1, 2]. A related class of proteins contains more subtle geometric configurations called slipknots [3, 4]. Recent theoretical studies using structure-based models (where native con- tacts are dominant) suggest that slipknot-like conforma- tions act like intermediates during the folding of knotted proteins [5]. This entire new mechanism is consistent with energy landscape theory (FEL) and the funnel con- cept [7, 8]. It was shown that the slipknot formation reduces the topological barrier. Complementing regular folding studies, additional information about the land- scape was obtained by mechanical manipulation of the knotted protein with atomic force microscopy [9] both experimentally in [10, 11] and theoretically in [12, 13, 14]. For example, [12] it has been showen that unfolding pro- ceeds via a series of jumps between various metastable conformations, a mechanism opposite to the smooth un- folding in knotted homopolymers. Most of our analysis is based on stretching simulations under constant force [16]. The crucial signature for this process is the overall unfolding time from the beginning of the stretching until the protein fully unfolds. Normally one expects that the transition between the native and the unfolded basins to be limited by overcoming the free energy barrier, which gets effectively reduced upon an application of a stretching force. The rate by which this barrier is reduced depends on the distance between the unfolded basin and the top of the barrier measured along the stretching coordinate x . This idea was first devel- oped in the phenomenological model of Bell [18], which states that the unfolding time τ decreases exponentially with applied stretching force F as τ ( F ) = τ$_{0}$e − Fx k B T . A Motivated by these early results, we now propose a uni- fied picture for the mechanical unfolding of proteins with slipknots. In this Letter this question is addressed by explaining the role of topological barriers along their me- chanical unfolding pathways. Supported by our previous results that knotted proteins can still have a minimally frustrated funnel-like energy landscape, structure-based theoretical coarse-grained models are used [15] to ana- lyze the behavior of a slipknot protein under stretching. Studies are performed for the α/β class protein thymi- FIG. 1: Dependence of the unfolding times τ on the stretch- ing force F for 1e2i (solid line, in red). In this Letter we describe this mechanism as a superposition of two unfolding pathways: I for small forces (dashed (lower) line, in blue), and II for intermidiate and large forces (dashed-dotted (up- per) line, green). arXiv:1001.0009v1[q-bio.BM]30 Dec 2009 2 3 4 5 F GLYPH<LBracket1> Ε GLYPH<Slash1> GLYPH<Angstrom> GLYPH<RBracket1> 7.2 7.5 7.8 8.1 log Τ