Learning Area Methodology B: Mathematics

Exam paper for second semester (Supplementary) Learning Area Methodology B: Mathematic

The Photosynthetic Response of Northern Red Oak (Quercus rubra L.) and American Chestnut (Castanea dentata (Marsh.) Borkh) Under Varying Light Intensity and Weed Competition

Although widely distributed across Indiana and the United States Midwest, conifer plantations consist largely of non-native species that are of no value to the state’s forest products industry. This project’s goal is to develop science-based protocols and specific silvicultural prescriptions for successfully converting conifer plantations to higher value native hardwoods. Quantifying photosynthesis rate in a plant is an important tool to help us discern the best methods for implementing conifer conversion. Seedlings from two different native species, northern red oak (Quercus rubra L.) and American chestnut (Castanea dentata (Marsh.) Borkh), were distributed among three different silvicultural cutting treatments (control, thinning and clear cut). Inside each one, two distinct categories of herbaceous control treatments (weed control and no weed control) were installed. Using an AccuPAR LP-80 sensor, canopy PAR (photosynthetically active radiation) interception was measured. Photosynthetic capacity was assessed with a LICOR 6400-XT analyzer to evaluate efficiency in resource use (water, light, gas exchange) and productivity. Among the treatments, clear cut presented maximum PAR intensity, followed by thinning and control, respectively. Both American chestnut and northern red oak seedlings demonstrated the highest photosynthesis rate (Amax) under high light conditions (clear cut), though photosynthesis of chestnut was greater than that of northern red oak. No significant differences were found between species in the weeding treatment for photosynthesis. Results of this study will provide valuable silvicultural prescriptions to Non-Industrial Private Forest (NIPF) landowners, forestry professionals, as well as state and federal agencies in Indiana and other Midwestern states

Calculating Ensemble Averaged Descriptions of Protein Rigidity without Sampling

Previous works have demonstrated that protein rigidity is related to thermodynamic stability, especially under conditions that favor formation of native structure. Mechanical network rigidity properties of a single conformation are efficiently calculated using the integer body-bar Pebble Game (PG) algorithm. However, thermodynamic properties require averaging over many samples from the ensemble of accessible conformations to accurately account for fluctuations in network topology. We have developed a mean field Virtual Pebble Game (VPG) that represents the ensemble of networks by a single effective network. That is, all possible number of distance constraints (or bars) that can form between a pair of rigid bodies is replaced by the average number. The resulting effective network is viewed as having weighted edges, where the weight of an edge quantifies its capacity to absorb degrees of freedom. The VPG is interpreted as a flow problem on this effective network, which eliminates the need to sample. Across a nonredundant dataset of 272 protein structures, we apply the VPG to proteins for the first time. Our results show numerically and visually that the rigidity characterizations of the VPG accurately reflect the ensemble averaged properties. This result positions the VPG as an efficient alternative to understand the mechanical role that chemical interactions play in maintaining protein stability

Hydrogen bond networks determine emergent mechanical and thermodynamic properties across a protein family

<p>Abstract</p> <p>Background</p> <p>Gram-negative bacteria use periplasmic-binding proteins (bPBP) to transport nutrients through the periplasm. Despite immense diversity within the recognized substrates, all members of the family share a common fold that includes two domains that are separated by a conserved hinge. The hinge allows the protein to cycle between open (apo) and closed (ligated) conformations. Conformational changes within the proteins depend on a complex interplay of mechanical and thermodynamic response, which is manifested as an increase in thermal stability and decrease of flexibility upon ligand binding.</p> <p>Results</p> <p>We use a distance constraint model (DCM) to quantify the give and take between thermodynamic stability and mechanical flexibility across the bPBP family. Quantitative stability/flexibility relationships (QSFR) are readily evaluated because the DCM links mechanical and thermodynamic properties. We have previously demonstrated that QSFR is moderately conserved across a mesophilic/thermophilic RNase H pair, whereas the observed variance indicated that different enthalpy-entropy mechanisms allow similar mechanical response at their respective melting temperatures. Our predictions of heat capacity and free energy show marked diversity across the bPBP family. While backbone flexibility metrics are mostly conserved, cooperativity correlation (long-range couplings) also demonstrate considerable amount of variation. Upon ligand removal, heat capacity, melting point, and mechanical rigidity are, as expected, lowered. Nevertheless, significant differences are found in molecular cooperativity correlations that can be explained by the detailed nature of the hydrogen bond network.</p> <p>Conclusion</p> <p>Non-trivial mechanical and thermodynamic variation across the family is explained by differences within the underlying H-bond networks. The mechanism is simple; variation within the H-bond networks result in altered mechanical linkage properties that directly affect intrinsic flexibility. Moreover, varying numbers of H-bonds and their strengths control the likelihood for energetic fluctuations as H-bonds break and reform, thus directly affecting thermodynamic properties. Consequently, these results demonstrate how unexpected large differences, especially within cooperativity correlation, emerge from subtle differences within the underlying H-bond network. This inference is consistent with well-known results that show allosteric response within a family generally varies significantly. Identifying the hydrogen bond network as a critical determining factor for these large variances may lead to new methods that can predict such effects.</p

Changes in Lysozyme Flexibility upon Mutation Are Frequent, Large and Long-Ranged

We investigate changes in human c-type lysozyme flexibility upon mutation via a Distance Constraint Model, which gives a statistical mechanical treatment of network rigidity. Specifically, two dynamical metrics are tracked. Changes in flexibility index quantify differences within backbone flexibility, whereas changes in the cooperativity correlation quantify differences within pairwise mechanical couplings. Regardless of metric, the same general conclusions are drawn. That is, small structural perturbations introduced by single point mutations have a frequent and pronounced affect on lysozyme flexibility that can extend over long distances. Specifically, an appreciable change occurs in backbone flexibility for 48% of the residues, and a change in cooperativity occurs in 42% of residue pairs. The average distance from mutation to a site with a change in flexibility is 17–20 Å. Interestingly, the frequency and scale of the changes within single point mutant structures are generally larger than those observed in the hen egg white lysozyme (HEWL) ortholog, which shares 61% sequence identity with human lysozyme. For example, point mutations often lead to substantial flexibility increases within the β-subdomain, which is consistent with experimental results indicating that it is the nucleation site for amyloid formation. However, β-subdomain flexibility within the human and HEWL orthologs is more similar despite the lowered sequence identity. These results suggest compensating mutations in HEWL reestablish desired properties