A target function for quaternary structural refinement from small angle scattering and NMR orientational restraints

Abstract

We present a novel target function based on atomic coordinates that permits quaternary structural refinement of multi-domain protein–protein or protein–RNA complexes. It requires that the high-resolution structures of the individual domains are known and that small angle scattering (SAS) data as well as NMR orientational restraints from residual dipolar couplings (RDCs) of the complex are available. We show that, when used in combination, the translational and rotational restraints contained in SAS intensities and RDCs, respectively, define a target potential function that permits to determine the overall topology of complexes made up of domains with low internal symmetry. We apply the target function on a modestly anisotropic model system, the Barnase/Barstar complex, and discuss factors that influence the structural refinement such as data errors and the geometrical properties of the individual domains.

Appendices

Appendix

Coefficients and partial derivatives of the B- and C-potentials are

$$ \begin{aligned} B(\theta, \phi) =& 4{\left({R_{1} + R_{2}} \right)}^{2} {\left[ \begin{aligned} \ & \sin^{2} \theta {\left({a_{\rm B} \cos ^{2} \phi + b_{\rm B} \sin ^{2} \phi + c_{\rm B} \sin \phi \cos \phi} \right)} + d_{\rm B} \cos ^{2} \theta \\ & + \sin \theta \cos \theta {\left({e_{\rm B} \cos \phi + f_{\rm B} \sin \phi} \right)} \\ \end{aligned} \right]} \\ & + 4{\left({R_{1} + R_{2}} \right)} {\left[ {\sin \theta {\left({g_{\rm B} \cos \phi + h_{\rm B} \sin \phi} \right)} + i_{\rm B} \cos \theta} \right]}, \\ \end{aligned}$$

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$$ \begin{aligned} a_{\rm B} =& {\sum\limits_{i \in K_{1}, j \in K_{2}} {{\left({x^{2}_{i} + x^{2}_{j}} \right)}},}\quad b_{\rm B} = {\sum\limits_{i \in K_{1}, j \in K_{2} } {{\left({y^{2}_{i} + y^{2}_{j}} \right),}}}\quad c_{\rm B} = 2{\sum\limits_{i \in K_{1} , j \in K_{2}} {{\left({x_{i} y_{i} + x_{j} y_{j}} \right),}}} \\ d_{\rm B} =& {\sum\limits_{i \in K_{1}, j \in K_{2}} {{\left({z^{2}_{i} + z^{2}_{j}} \right)}}} ,\quad e_{\rm B} = 2{\sum\limits_{i \in K_{1}, j \in K_{2}} {{\left({x_{i} z_{i} + x_{j} z_{j}} \right),}}}\quad f_{\rm B} = 2{\sum\limits_{i \in K_{1}, j \in K_{2}} {{\left({y_{i} z_{i} + y_{j} z_{j}} \right)}}} ,\\ g_{\rm B} =& {\sum\limits_{i \in K_{1}, j \in K_{2}} {{\left({x^{3}_{i} + y^{2}_{i} x_{i} + z^{2}_{i} x_{i} - x^{3}_{j} - y^{2}_{j} x_{j} - z^{2}_{j} x_{j}} \right)}}} ,\\ h_{\rm B} =& {\sum\limits_{i \in K_{1}, j \in K_{2}} {{\left({x^{2}_{i} y_{i} + y^{3}_{i} + z^{2}_{i} y_{i} - x^{2}_{j} y_{j} - y^{3}_{j} - z^{2}_{j} y_{j}} \right)}}} ,\\ i_{\rm B} =& {\sum\limits_{i \in K_{1}, j \in K_{2} } {{\left({x^{2}_{i} z_{i} + y^{2}_{i} z_{i} + z^{3}_{i} - x^{2}_{j} z_{j} - y^{2}_{j} z_{j} - z^{3}_{j}} \right)}}} ,\\ \end{aligned} $$

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$$ \begin{aligned} \frac{\partial}{{\partial \phi}}B(\theta, \phi) =& 4{\left({R_{1} + R_{2}} \right)}^{2} {\left[ \begin{aligned}\ & \sin ^{2} \theta {\left({- 2a_{\rm B} \cos \phi \sin \phi + 2b_{\rm B} \sin \phi \cos \phi + c_{\rm B} {\left({\cos ^{2} \phi - \sin ^{2} \phi} \right)}} \right)} \\ & + \sin \theta \cos \theta {\left({- e_{\rm B} \sin \phi + f_{\rm B} \cos \phi} \right)} \\ \end{aligned} \right]} \\ &+ 4{\left({R_{1} + R_{2}} \right)} {\left[ {\sin \theta {\left({- g_{\rm B} \sin \phi + h_{\rm B} \cos \phi} \right)}} \right]} ,\\ \frac{\partial}{{\partial \theta}}B(\theta, \phi) =& 4{\left({R_{1} + R_{2}} \right)}^{2} {\left[ \begin{aligned}\ & 2\sin \theta \cos \theta {\left({a_{\rm B} \cos ^{2} \phi + b_{\rm B} \sin ^{2} \phi + c_{\rm B} \sin \phi \cos \phi} \right)} \\ & - 2d_{\rm B} \cos \theta \sin \theta + {\left({\cos ^{2} \theta - \sin ^{2} \theta} \right)}{\left({e_{\rm B} \cos \phi + f_{\rm B} \sin \phi} \right)} \\ \end{aligned} \right]} \\ &+ 4{\left({R_{1} + R_{2}} \right)} {\left[ {\cos \theta {\left({g_{\rm B} \cos \phi + h_{\rm B} \sin \phi} \right)} - i_{\rm B} \sin \theta} \right]}, \\ \end{aligned}$$

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$$ \begin{aligned} & C(\theta, \phi) = 24{\left({R_{1} + R_{2}} \right)}^{4} {\left[ \begin{aligned} \ & \sin ^{2} \theta {\left({a_{\rm C} \cos ^{2} \phi + b_{\rm C} \sin ^{2} \phi + c_{\rm C} \sin \phi \cos \phi} \right)} + d_{\rm C} \cos ^{2} \theta \\ & + \sin \theta \cos \theta {\left({e_{\rm C} \cos \phi + f_{\rm C} \sin \phi} \right)} \\ \end{aligned} \right]} \\ & \quad \quad \quad\,\,+ 24{\left({R_{1} + R_{2}} \right)}^{3} {\left[ {\sin \theta {\left({g_{\rm C} \cos \phi + h_{\rm C} \sin \phi} \right)} + i_{\rm C} \cos \theta} \right]} \\ & \quad \quad \quad \; + 16{\left({R_{1} + R_{2}} \right)}^{3} {\left[ \begin{aligned}\ & \sin ^{3} \theta {\left({j_{\rm C} \cos ^{3} \phi + k_{\rm C} \sin ^{3} \phi + l_{\rm C} \sin \phi \cos ^{2} \phi + m_{\rm C} \sin ^{2} \phi \cos \phi} \right)} \\ & + n_{\rm C} \cos ^{3} \theta + \sin ^{2} \theta \cos \theta {\left({o_{\rm C} \cos ^{2} \phi + p_{\rm C} \sin ^{2} \phi + q_{\rm C} \cos \phi \sin \phi} \right)} \\ & + \sin \theta \cos ^{2} \theta {\left({r_{\rm C} \cos \phi + s_{\rm C} \sin \phi} \right)} \\ \end{aligned} \right]} ,\\ \end{aligned} $$

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$$ \begin{aligned} a_{\rm C} =& a_{\rm B} + {\sum\limits_{i \in K_{1},j \in K_{2}} {{\left({x^{2}_{i} + x^{2}_{j}} \right)}\frac{{r^{2}_{{ij}}}}{{{\left({R_{1} + R_{2}} \right)}^{2}}}},}\quad b_{\rm C} = b_{\rm B} + {\sum\limits_{i \in K_{1}, j \in K_{2}} {{\left({y^{2}_{i} + y^{2}_{j}} \right)}\frac{{r^{2}_{{ij}}}}{{{\left({R_{1} + R_{2}} \right)}^{2}}}}} ,\\ c_{\rm C} =& c_{\rm B} + 2{\sum\limits_{i \in K_{1}, j \in K_{2}} {{\left({x_{i} y_{i} + x_{j} y_{j}} \right)}}}\frac{{r^{2}_{{ij}}}}{{{\left({R_{1} + R_{2}} \right)}^{2}},}\quad d_{\rm C} = d_{\rm B} + {\sum\limits_{i \in K_{1}, j \in K_{2}} {{\left({z^{2}_{i} + z^{2}_{j}} \right)}\frac{{r^{2}_{{ij}}}}{{{\left({R_{1} + R_{2}} \right)}^{2}}}}} ,\\ e_{\rm C} =& e_{\rm B} + {\sum\limits_{i \in K_{1}, j \in K_{2}} {{\left({x_{i} z_{i} + x_{j} z_{j}} \right)}\frac{{r^{2}_{{ij}}}}{{{\left({R_{1} + R_{2}} \right)}^{2}}}},}\quad f_{\rm C} = f_{\rm B} + {\sum\limits_{i \in K_{1}, j \in K_{2}} {{\left({y_{i} z_{i} + y_{j} z_{j}} \right)}\frac{{r^{2}_{{ij}}}}{{{\left({R_{1} + R_{2}} \right)}^{2}}}}} ,\\ g_{\rm C} =& g_{\rm B} + {\sum\limits_{i \in K_{1}, j \in K_{2}} {{\left({x^{3}_{i} + y^{2}_{i} x_{i} + z^{2}_{i} x_{i} - x^{3}_{j} - y^{2}_{j} x_{j} - z^{2}_{j} x_{j}} \right)}}}\frac{1}{2}\frac{{r^{2}_{{ij}}}}{{{\left({R_{1} + R_{2}} \right)}^{2}}} ,\\ h_{\rm C} =& h_{\rm B} + {\sum\limits_{i \in K_{1}, j \in K_{2}} {{\left({x^{2}_{i} y_{i} + y^{3}_{i} + z^{2}_{i} y_{i} - x^{2}_{j} y_{j} - y^{3}_{j} z^{2}_{j} y_{j}} \right)}}}\frac{1}{2}\frac{{r^{2}_{{ij}}}}{{{\left({R_{1} + R_{2}} \right)}^{2}}}, \\ i_{\rm C} =& i_{\rm B} + {\sum\limits_{i \in K_{1}, j \in K_{2}} {{\left({x^{2}_{i} z_{i} + y^{2}_{i} z_{i} + z^{3}_{i} - x^{2}_{j} z_{j} - y^{2}_{j} z_{j} - z^{3}_{j}} \right)}}}\frac{1}{2}\frac{{r^{2}_{{ij}}}}{{{\left({R_{1} + R_{2}} \right)}^{2}}} ,\\ j_{\rm C} =& {\sum\limits_{i \in K_{1}, j \in K_{2}} {{\left({x^{3}_{i} - x^{3}_{j}} \right)}},}\quad k_{\rm C} = {\sum\limits_{ i \in K_{1}, j \in K_{2}} {{\left({y^{3}_{i} - y^{3}_{j}} \right)}},}\quad l_{\rm C} = 3{\sum\limits_{i \in K_{1}, j \in K_{2}} {{\left({x^{2}_{i} y_{i} - x^{2}_{j} y_{j}} \right)}},} \\ m_{\rm C} =& 3{\sum\limits_{i \in K_{1}, j \in K_{2}} {{\left({y^{2}_{i} x_{i}-y^{2}_{j} x_{j}} \right)}}} ,\quad n_{\rm C} = {\sum\limits_{i \in K_{1}, j \in K_{2}} {{\left({z^{3}_{i} - z^{3}_{j}} \right)}},}\quad o_{\rm C} = 3{\sum\limits_{i \in K_{1}, j \in K_{2}} {{\left({x^{2}_{i} z_{i} - x^{2}_{j} z_{j}} \right)}},}\\ p_{\rm C} =& 3{\sum\limits_{i \in K_{1}, j \in K_{2}} {{\left({y^{2}_{i} z_{i} - y^{2}_{j} z_{j}} \right)}}}, \quad q_{\rm C} = 6{\sum\limits_{i \in K_{1}, j \in K_{2}} {{\left({x_{i} y_{i} z_{i} - x_{j} y_{j} z_{j}} \right)}}} , \quad r_{C} = 3{\sum\limits_{i \in K_{1}, j \in K_{2}} {{\left({z^{2}_{i} x_{i} - z^{2}_{j} x_{j}} \right)}},}\\ s_{\rm C} =& 3{\sum\limits_{i \in K_{1}, j \in K_{2}} {{\left({z^{2}_{i} y_{i} - z^{2}_{j} y_{j}} \right)}}}, \\ \end{aligned} $$

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$$ \begin{aligned} \frac{\partial}{{\partial \phi}}C(\theta, \phi) =& 24{\left({R_{1} + R_{2}} \right)}^{4} {\left[ \begin{aligned} \ & \sin ^{2} \theta {\left({- 2a_{\rm C} \cos \phi \sin \phi + 2b_{\rm C} \sin \phi \cos \phi + c_{\rm C} {\left({- \sin ^{2} \phi + \cos ^{2} \phi} \right)}} \right)} \\ & + \sin \theta \cos \theta {\left({- e_{\rm C} \sin \phi + f_{\rm C} \cos \phi} \right)} \\ \end{aligned} \right]} \\ & + 24{\left({R_{1} + R_{2}} \right)}^{3} {\left[ {\sin \theta {\left({- g_{\rm C} \sin \phi + h_{\rm C} \cos \phi} \right)}} \right]} \\ & + 16{\left({R_{1} + R_{2}} \right)}^{3} {\left[ \begin{aligned}\ & \sin ^{3} \theta {\left(\begin{aligned}\ & - 3j_{\rm C} \cos ^{2} \phi \sin \phi + 3k_{\rm C} \sin ^{2} \phi \cos \phi \\ & + l_{\rm C} {\left({\cos ^{3} \phi - 2 \sin ^{2} \phi \cos \phi} \right)} + m_{\rm C} {\left({- \sin ^{3} \phi + 2\sin \phi \cos ^{2} \phi} \right)} \\ \end{aligned} \right)} \\ & + \sin ^{2} \theta \cos \theta {\left(\begin{aligned}\ & - 2o_{\rm C} \cos \phi \sin \phi + 2p_{\rm C} \sin \phi \cos \phi \\ & + q_{\rm C} {\left({- \sin ^{2} \phi + \cos ^{2} \phi} \right)} \\ \end{aligned} \right)} \\ & + \sin \theta \cos ^{2} \theta {\left({- r_{\rm C} \sin \phi + s_{\rm C} \cos \phi} \right)} \\ \end{aligned} \right]} , \\ \frac{\partial}{{\partial \theta}}C(\theta, \phi) =& 24{\left({R_{1} + R_{2}} \right)}^{4} {\left[ \begin{aligned}\ & 2\sin \theta \cos \theta {\left({a_{\rm C} \cos ^{2} \phi + b_{\rm C} \sin ^{2} \phi + c_{\rm C} \sin \phi \cos \phi} \right)} - 2d_{\rm C} \cos \theta \sin \theta \\ & + {\left({- \sin ^{2} \theta + \cos ^{2} \theta} \right)}{\left({e_{\rm C} \cos \phi + f_{\rm C} \sin \phi} \right)} \\ \end{aligned} \right]} \\ &+ 24{\left({R_{1} + R_{2}} \right)}^{3}{\left[ {\cos \theta {\left({g_{\rm C} \cos \phi + h_{\rm C} \sin \phi} \right)} - i_{\rm C} \sin \theta} \right]}\\ & + 16{\left({R_{1} + R_{2}} \right)}^{3} {\left[ \begin{aligned}\ & 3\sin ^{2} \theta \cos \theta {\left({j_{\rm C} \cos ^{3} \phi + k_{\rm C} \sin ^{3} \phi + l_{\rm C} \sin \phi \cos ^{2} \phi + m_{\rm C} \sin ^{2} \phi \cos \phi} \right)} \\ & - 3n_{\rm C} \cos ^{2} \theta \sin \theta \\ & + {\left({\sin ^{3} \theta + 2\sin \theta \cos ^{2} \theta} \right)}{\left({o_{\rm C} \cos ^{2} \phi + p_{\rm C} \sin ^{2} \phi + q_{\rm C} \cos \phi \sin \phi} \right)} \\ & + {\left({\cos ^{3} \theta - 2\sin ^{2} \theta \cos \theta} \right)}{\left({r_{\rm C} \cos \phi + s_{\rm C} \sin \phi} \right)} \\ \end{aligned} \right]}. \\ \end{aligned} $$

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Distribution of initial and refined Barstar positions

See Fig. 9.

Fig. 9

Distribution of the initial and of the refined end position coordinates (θ, ϕ ) of the Barstar domain. The initial positions are depicted in black, structures that converged at less than 1,000 refinement steps in green, and structures that had not converged after 1,000 steps are depicted in red