On Provable Algorithms for Determination of Continuous Protein Interdomain Motions from Residual Dipolar Couplings

Dynamics of biomolecules over various spatial and time scales are essential for biological functions such as molecular recognition, catalysis and signaling. However, reconstruction of biomolecular dynamics from experimental observables requires the determination of a conformational probability distribution. Unfortunately, these distributions cannot be fully constrained by the limited information from experiments, making the problem an ill-posed one in the terminology of Hadamard. The ill-posed nature of the problem comes from the fact that it has no unique solution. Multiple or even an infinite number of solutions may exist. To avoid the ill-posed nature, the problem needs to be regularized by making assumptions, which inevitably introduce biases into the result.

Here, I present two continuous probability density function approaches to solve an important inverse problem called the RDC trigonometric moment problem. By focusing on interdomain orientations we reduced the problem to determination of a distribution on the 3D rotational space from residual dipolar couplings (RDCs). We derived an analytical equation that relates alignment tensors of adjacent domains, which serves as the foundation of the two methods. In the first approach, the ill-posed nature of the problem was avoided by introducing a continuous distribution model, which enjoys a smoothness assumption. To find the optimal solution for the distribution, we also designed an efficient branch-and-bound algorithm that exploits the mathematical structure of the analytical solutions. The algorithm is guaranteed to find the distribution that best satisfies the analytical relationship. We observed good performance of the method when tested under various levels of experimental noise and when applied to two protein systems. The second approach avoids the use of any model by employing maximum entropy principles. This 'model-free' approach delivers the least biased result which presents our state of knowledge. In this approach, the solution is an exponential function of Lagrange multipliers. To determine the multipliers, a convex objective function is constructed. Consequently, the maximum entropy solution can be found easily by gradient descent methods. Both algorithms can be applied to biomolecular RDC data in general, including data from RNA and DNA molecules.

Contributors to and impact of residual shunting after device closure of atrial septal defects.

BACKGROUND: The prevalence of residual shunt in patients after device closure of atrial septal defect and its impact on long-term outcome has not been previously defined. METHODS: From a prospective, single-institution registry of 408 patients, we selected individuals with agitated saline studies performed 1 year after closure. Baseline echocardiographic, invasive hemodynamic, and comorbidity data were compared to identify contributors to residual shunt. Survival was determined by review of the medical records and the Social Security Death Index. Survival analysis according to shunt included construction of Kaplan-Meier curves and Cox proportional hazards modeling. RESULTS: Among 213 analyzed patients, 27% were men and age at repair was 47 ± 17 years. Thirty patients (14%) had residual shunt at 1 year. Residual shunt was more common with Helex (22%) and CardioSEAL/STARFlex (40%) occluder devices than Amplatzer devices (9%; P = .005). Residual shunts were more common in whites (79% vs 46%, P = .004). At 7.3 ± 3.3 years of follow-up, 13 (6%) of patients had died, including 8 (5%) with Amplatzer, 5 (25%) with CardioSEAL/STARFlex, and 0 with Helex devices. Patients with residual shunting had a higher hazard of death (20% vs 4%, P = .001; hazard ratio 4.95 [1.59-14.90]). In an exploratory multivariable analysis, residual shunting, age, hypertension, coronary artery disease, and diastolic dysfunction were associated with death. CONCLUSIONS: Residual shunt after atrial septal defect device closure is common and adversely impacts long-term survival.