Stochastic processes with optimization — A model of learning in higher systems

A special class of stochastic processes with optimization (SPO) is considered and their long-run behaviour is investigated. At each step of the process {X _h} _h 0 (where X _h is a discrete random variable) a loss function expressing the distance with respect to the moments in the previous step is minimized. The transformation leading from a certain probability distribution F _k (step k) to the next probability distribution F _k+1 (step k+1) is accomplished by means of an optimization operator, or simply optimator, that is, a nonlinear operator performing an optimization. The constraints involved by the optimator are typically regarded as a message conveying the information needed for the stepwise evolution of the system. In other words, the behaviour of the system is expressed by an ordered set of events (actions) to be realized with given probabilities and minimum losses, while the message is viewed as a set of constraints providing an inferior bound for that probabilities, hence, ensuring that the required actions are performed. Besides, in order to account for some relaxation phenomena taking place in higher systems each active step (active optimator) is followed by a relaxation step (recovery optimator). Under these conditions it is shown that repeated presentation of a stimulus pattern leads to a convergent process, so that the action is finally performed with minimum minimorum losses. This reveals some fundamental relations between optimization and learning in higher systems, highlighting also the key role of the relaxation processes. Further the behaviour of the system is described in terms of a special class of Non-Markovian processes termed stochastic processes with optimization and relaxation (SPORs). It is shown that two basic subclasses of SPORs exist, namely the monoergodic and the biergodic SPORs. Sufficient conditions for both monoergodicity and biergodicity are given. Finally, a particular feature of the optimators, the so-called nonredundancy is shown to be relevant with respect to the influence of the past on the current evolution of the system.     

Prediction of protein secondary structure from amino acid sequence

The conformational parametersP _k for each amino acid species (j=1–20) of sequential peptides in proteins are presented as the product ofP _i,k , wherei is the number of the sequential residues in thekth conformational state (k=-helix,-sheet,-turn, or unordered structure). Since the average parameter for ann-residue segment is related to the average probability of finding the segment in the kth state, it becomes a geometric mean of (P _k )_av=(P _i,k ) ^1/n with amino acid residuei increasing from 1 ton. We then used ln(P_k)_av to convert a multiplicative process to a summation, i.e., ln(P _k ) _av =(1/n)P _i,k (i=1 ton) for ease of operation. However, this is unlike the popular Chou-Fasman algorithm, which has the flaw of using the arithmetic mean for relative probabilities. The Chou-Fasman algorithm happens to be close to our calculations in many cases mainly because the difference between theirP _k and our InP _k is nearly constant for about one-half of the 20 amino acids. When stronger conformation formers and breakers exist, the difference become larger and the prediction at the N- and C-terminal-helix or-sheet could differ. If the average conformational parameters of the overlapping segments of any two states are too close for a unique solution, our calculations could lead to a different prediction.     

Minimal entropy probability paths between genome families

We develop a metric for probability distributions with applications to biological sequence analysis. Our distance metric is obtained by minimizing a functional defined on the class of paths over probability measures on N categories. The underlying mathematical theory is connected to a constrained problem in the calculus of variations. The solution presented is a numerical solution, which approximates the true solution in a set of cases called rich paths where none of the components of the path is zero. The functional to be minimized is motivated by entropy considerations, reflecting the idea that nature might efficiently carry out mutations of genome sequences in such a way that the increase in entropy involved in transformation is as small as possible. We characterize sequences by frequency profiles or probability vectors, in the case of DNA where N is 4 and the components of the probability vector are the frequency of occurrence of each of the bases A, C, G and T. Given two probability vectors a and b, we define a distance function based as the infimum of path integrals of the entropy function H(p) over all admissible paths p(t), 0 t1, with p(t) a probability vector such that p(0)=a and p(1)=b. If the probability paths p(t) are parameterized as y(s) in terms of arc length s and the optimal path is smooth with arc length L, then smooth and rich optimal probability paths may be numerically estimated by a hybrid method of iterating Newtons method on solutions of a two point boundary value problem, with unknown distance L between the abscissas, for the Euler–Lagrange equations resulting from a multiplier rule for the constrained optimization problem together with linear regression to improve the arc length estimate L. Matlab code for these numerical methods is provided which works only for rich optimal probability vectors. These methods motivate a definition of an elementary distance function which is easier and faster to calculate, works on non–rich vectors, does not involve variational theory and does not involve differential equations, but is a better approximation of the minimal entropy path distance than the distance ||b–a||₂. We compute minimal entropy distance matrices for examples of DNA myostatin genes and amino-acid sequences across several species. Output tree dendograms for our minimal entropy metric are compared with dendograms based on BLAST and BLAST identity scores.Mathematics Subject Classification (2000): 92B05, 92D20     

Bubble coalescence behaviour in biological media

Summary Volumetric mass transfer coefficients (k_La) were measured by a steady state method in a twin bubble column to characterize the coalescence behaviour of the medium. Employing Hansenula polymorpha cultivation broths, k_La values were compared with those measured in model media in the presence and absence of antifoam agents. The ratio of the volumetric mass transfer coefficient in the system investigated to that in water, , was employed to characterize the cultivation medium.Symbols a Specific gas/liquid interfacial area with regard to the liquid volume in reactor - d_e Dynamical equilibrium bubble diameter - d_H Perforated plate hole diameter - d_p Primary bubble diameter - d_S Sauter bubble diameter - F_v Liquid feed rate - H Bubbling layer height - k_L Gas/liquid mass transfer coefficient - k_La Volumetric mass transfer coefficient - m k_La/(k_La)_r coalescence index - m_corr Corrected coalescence index [Eq. (3)] - OTR Oxygen transfer rate - P_O Dissolved O₂-partial pressure in BS2 - P₁ Dissolved O₂-partial pressure in BS1 - P_O P_O/P_S relative oxygen saturation in BS2 - P₁ P₁/P_S relative oxygen saturation in BS1 - P_S Saturation dissolved oxygen partial pressure - R_c dn_B/dt coalescence rate - S Substrate concentration - t_F Time since the beginning of the cultivation - X Biomass concentration - V₁ Liquid volume in BS1 - w_SG Superficial gas velocity in BS1 - G Gas holdup in BS1 - ₁ V₁/F_v mean liquid residence time in BS1 - BS1 O₂ absorber column - BS2 O₂ desorber column - D Desmophen (antifoam agent) - NS Nutrient salt solution (Table 1)     

Joint Confidence Sets in Multiple Dilution Assays

Let {Q₁, …, Q_k} be the potencies of k substances relative to a standard in a multiple dilution assay. Joint confidence bounds for these are given with confidence coefficient at least 1-α. These bounds are easily interpreted; they appeal to available tables; they improve Scheffe's bounds; they are based on applicable probability inequalities together with extensions of Fieller's theorem; and they are genuinely nonparametric. The procedures are illustrated using data from parallel-line and slope-ratio assays.     

Fixation of beneficial mutations in the presence of epistatic interactions

We investigate the effect of deleterious mutations on the process of fixation of new advantageous mutants in an asexual population. In particular we wish to study the dependence of the process on the strength of the deleterious mutations. We suppose the existence of epistatic interaction between the genes. We study the model by means of branching process theory and also by numerical simulations. Our results show the occurrence of two distinct regimes of behavior for the probability of fixation of these variants. The occurrence of either regime depends on the ratio between the selective advantage of the beneficial mutation s _b and on the selective parameter for deleterious mutations s _b . In the former, which takes place for s _b /s _d ≲ 1, the probability of fixation increases with the epistasis parameter α, whereas for s _b /s _d ≫ 1 the probability of fixation is a complex function of α and the mutation rate U. Surprisingly, we find that for the multiplicative landscape (α = 1) the probability of fixation P _fix is given by where π (s _b ) is the probability of fixation for the two-allele model in the absence of mutations as calculated by Haldane (1927, Proc. Camb. Phil. Soc., 26, 220–230) and Kimura (1962, Genetics, 47, 713–719).     

Phosphomolybdate cluster based solids mediated by transition metal complexes

Reaction of molybdate and phosphate precursors in the presence of 3d transition metal ions and pyrazole (pz) under hydrothermal condition resulted in the crystallization of four new phosphomolybdate cluster based solids: (pz)₂[{Co(pz)₄}₅{P₂Mo₅O₂₃}₂]·6H₂O (1), (pz){Ni(pz)₄(H₂O)₂}[{Ni(pz)₄}₅{P₂Mo₅O₂₃}₂]·2H₂O (2), {Cu(pz)₄(H₂O)₂}[{Cu(pz)₄}{Cu(pz)₄(H₂O)}{P₂Mo₅O₂₃}]·2H₂O (3) and (pz)[{Zn(pz)₃}₃{P₂Mo₅O₂₃}]·2H₂O (4). In all the solids, a metal complex {M(pz)_n} covalently links{P₂Mo₅O₂₃} clusters to form a chain. The dimensionality of the structures differs in the way the chains link or self assemble with counter ions and water molecules. To the best of our knowledge, 4 is the first example of a zinc complex incorporated with {P₂Mo₅O₂₃} cluster. The paper discusses the self assembly occurring between in situ metal pyrazole complex and phosphomolybdate clusters through coordinate/covalent and noncovalent interactions during crystallization of a particular solid.     

Reduced order multiport parallel and multidirectional neural associative memories

This paper proposes multiport parallel and multidirectional intraconnected associative memories of outer product type with reduced interconnections. Some new reduced order memory architectures such as k-directional and k-port parallel memories are suggested. These architectures are, also, very suitable for implementation of spatio-temporal sequences and multiassociative memories. It is shown that in the proposed memory architectures, a substational reduction in interconnections is achieved if the actual length of original N-bit long vectors is subdivided into k sublengths. Using these sublengths, submemory matrices, T _s or W _s , are computed, which are then intraconnected to form k-port parallel or k-directional memories. The subdivisions of N-bit long vectors into k sublengths save of interconnections. It is shown, by means of an example, that more than 80% reduction in interconnections is achieved. Minimum limit in bits on k as well as maximum limit on subdivisions in k is determined. The topologies of reduced interconnectivity developed in this paper are symmetric in structure and can be used to scale up to larger systems. The underlying principal of construction, storage and retrieval processes of such associative memories has been analyzed. The effect of complexity of different levels of reduced interconnectivity on the quality of retrieval, signal to noise ratio, and storage capacity has been investigated. The model possesses analogies to biological neural structures and digital parallel port memories commonly used in parallel and multiprocessing systems.     

Enhanced oxygen transfer rates in fermentation using soybean oil-in-water dispersions

Summary The effect of soybean oil on the volumetric oxygen transfer coefficient during the cultivation ofAerobacter aerogenes cells is presented. For our aeration-agitation conditions (0.278 vvm and 500 rpm), it has been demonstrated that the use 19% (v/v) of soybean oil enabled a 1.85-fold increase of thek _l a coefficient (calculated on a per liter aqueous phase basis). For smaller volumetric oil fractions,k _L a increased linearly with the oil loading. Because of the oxygen-vector properties of soybean oil, this oil is able to significantly increase thek _L a of a bioreactor.Nomenclature C^*, C saturation and actual dissolved oxygen concentrations respectively (g/m³) - K_La volumetric oxygen transfer coefficient (h^–1) - K_La_initial k _La measured before the oil addition (h^–1) - M_O2 molar mass of oxygen (dalton) - N oxygen transfer rate (g/m³. h) - P_O2. P_N2 partial pressures ofO ₂ andN ₂ in the gas (atm) - P_H2OT partial pressure of water in air at the temperatureT (atm) - P_T total pressure (atm) - Q₀ volumetric flow rate of outlet air before seeding (m³/h) - Sp spreading coefficient (dynes/cm) - T absolute temperature of outlet gas (K) - V_i volume of the liquidi in the fermentor (m³) - V_M molar volume at 273 K and 1 atm (m³/mole) - _ij interfacial tension betweeni andj componants (dynes/cm) - _v volumetric fraction of the oil (v/v) - G gas - O oil - W water - i inlet - o outlet     

Synthesis of iron thienyl complexes

Dimetallation of thiophene (TH₂), bithiophene (BTH₂) and 3,6-dimethyl[3,2-b]thienothiophene (TTH₂) using a slight excess of butyl lithium, followed by the addition of [FeCp(CO)₂I], resulted in the formation of [2,5-{FeCp(CO)₂}₂T], 1 and [2-{FeCp(CO)₂}T]. The analogous reaction with bithiophene as precursor afforded similar products [2,2′-{FeCp(CO)₂}₂BT] 2 and [2-{FeCp(CO)₂}BTH] 3. In addition to the expected mono- ([2-{FeCp(CO)₂}-TTH] 4) and binuclear ([2,2′-{FeCp(CO)₂}₂-TT] 5) products, dimetallation of 3,6-dimethyl[3,2-b]thienothiophene and the subsequent reaction with [FeCp(CO)₂I] yielded carbonyl inserted mono-([2-{FeCp(CO)₂}C(O)-{TT}₂H] 6) and binuclear ([2-{FeCp(CO)₂}C(O)-{TT}₂-2′-{FeCp(CO)₂}] 7) carbon-carbon coupled products. The precursor [2,7-{SnMe₃}₂-TT] (8) was prepared and reacted with [FeCp(CO)(PEt₃)I] in the presence of a palladium catalyst to afford [2-{FeCp(CO)(PEt₃)}C(O)-{TT}₂-2′-{SnMe₃}] 10.     

The effects of fermentation conditions on yeast cell debris particle size distribution during high pressure homogenisation

Experiments were performed to characterize the particle size distribution of bakers' yeast cells during high pressure homogenisation. Results were obtained for mechanically agitated batch and continuously grown cultures under a range of operating conditions. It was found that the dependency of cell debris size distribution on the number of passes through the homogeniser and the homogeniser pressure was independent of the cell properties and culture conditions, but for a fixed pressure and number of passes the extent of disruption was strongly affected by the operating conditions in the fermenter. The entire cell debris size distributions were successfully simulated using the mean and variance of the distributions and a previously published model equation which related these parameters to the operating pressure and number of passes through the homogeniser.List of Symbols k breakage coefficient in Eq. 1 - d cell diameter - d ₅₀ median diameter of homogenate size distribution - d ₅₀ dimensionless d ₅₀ defined as - D dilution rate - F(d _NP) cumulative undersize distribution (volume basis) - N number of passes - P total pressure - P _threshold threshold pressure - P (P-P _threshold) - w Boltzmann parameter, Eq. 4 - w dimensionless standard deviation defined as Greek Letters exponent in Eq. 1 - exponent in Eq. 1 UCL is the Biotechnology and Biological Sciences Research Council's Interdisciplinary Research Centre for Biochemical Engineering and the Council's support to the participating UCL departments is gratefully acknowledged. The provision of continuous fermentation material from Dr. M. Gregory, Process System Engineering IRC, is gratefully acknowledged.     

StreAM-$$T_g$$: algorithms for analyzing coarse grained RNA dynamics based on Markov models of connectivity-graphs

Background

In this work, we present a new coarse grained representation of RNA dynamics. It is based on adjacency matrices and their interactions patterns obtained from molecular dynamics simulations. RNA molecules are well-suited for this representation due to their composition which is mainly modular and assessable by the secondary structure alone. These interactions can be represented as adjacency matrices of k nucleotides. Based on those, we define transitions between states as changes in the adjacency matrices which form Markovian dynamics. The intense computational demand for deriving the transition probability matrices prompted us to develop StreAM-\(T_g\), a stream-based algorithm for generating such Markov models of k-vertex adjacency matrices representing the RNA.

Results

We benchmark StreAM-\(T_g\) (a) for random and RNA unit sphere dynamic graphs (b) for the robustness of our method against different parameters. Moreover, we address a riboswitch design problem by applying StreAM-\(T_g\) on six long term molecular dynamics simulation of a synthetic tetracycline dependent riboswitch (500 ns) in combination with five different antibiotics.

Conclusions

The proposed algorithm performs well on large simulated as well as real world dynamic graphs. Additionally, StreAM-\(T_g\) provides insights into nucleotide based RNA dynamics in comparison to conventional metrics like the root-mean square fluctuation. In the light of experimental data our results show important design opportunities for the riboswitch.

     

Nickel promoted C-H, C-C and C-O bond activation in solution

Reaction of NiI₂ with the PCP-ligand {1-Et-2,6-(CH₂PⁱPr₂)₂-C₆H₃} (1) results in selective activation of the strong sp²-sp³ aryl-ethyl bond to afford the aryl-nickel complex [Ni{2,6-(CH₂PⁱPr₂)₂-C₆H₃}I] (2), whereas reaction of NiI₂ with {1,3,5-(CH₃)₃-2,6-(CH₂PⁱPr₂)₂-C₆H} (4) leads to the formation of the benzylic complex [Ni{1-CH₂-2,6-(CH₂PⁱPr₂)₂-3,5-(CH₃)₂-C₆H}I] (5) by selective C-H bond activation. Thermolysis of 5 results in formation of [Ni{2,6-(CH₂PⁱPr₂)₂-3,5-(CH₃)₂-C₆H}I] (6) by activation of the sp²-sp³ C-C bond. The identity of the new 16-electron complexes 2 and 6 was confirmed by reaction of NiI₂ with {1,3-(CH₂PⁱPr₂)₂-C₆H₄} (3) and {1,3-(CH₃)₂-4,6-(CH₂PⁱPr₂)₂-C₆H₂} (7), respectively, lacking the aryl-alkyl groups between the “phosphines arms” (alkyl=ethyl, methyl). Complexes 2 and 5 have been fully characterized by X-ray analysis. Nickel-based activation of an unstrained C-O single bond was observed as well. Reaction of the aryl-methoxy bisphosphine {1-OMe-2,6-(CH₂PⁱPr₂)₂-C₆H₃} (8) with NiI₂ results in the formation of the phenoxy complex [Ni{1-O-2,6-(CH₂PⁱPr₂)₂-C₆H₃}I] (9) by selective sp³-sp³ C-O bond activation.     

Monovalent cation permeabilities of the potassium systems in the crab giant axon

Summary Permeability ratios for pairs of monovalent cations permeating the two potassium systems proposed for the giant axon of the crabCarcinus maenas (M. E. Quinta-Ferreira, E. Rojas & N. Arispe,J. Membrane Biol. 66:171–181, 1982b) were estimated from measurements of the reversal potential of the currents under voltage-clamp conditions. With K⁺ inside the axon, permeability ratios from the reversal potential of the currents through the late channel are:P _Rb/P _K=0.9, /P _K<0.2 andP _Cs/P _K=0.18. With Cs⁺ inside the ratios are:P _K/P _Cs=8.7,P _Rb/P _Cs=7.1 and /P _Cs=2.4. The analysis of the inward currents carried by Rb⁺ or NH ₄ ⁺ showed similar reversal potentials for the early transient component and the late sustained component. Whence, the sequence of permeabilities for the two types of potassium channels is:P _K>P _Rb> >P _Na=P _Cs. The time constants for the activation of the two components recorded either in K-, Rb-, or NH₄-artificial seawater are twice as large as the corresponding time constants measured in Na-artificial seawater.     

Rates of decay for the survival probability of a mutant gene

If qk is the extinction probability of a slightly supercritical branching process with offspring distribution P _kr : r = 0, 1, 2,..., then it is shown that if sup _r r ³ p _kr , < , inf ² _k > 0, and m _k 1, then 1 – q _k 2(m _k –1) _k ^–2, where m _k = _r rp _kr , ² _k = ^k _r r ² p _kr – m _k ². This provides a simple set of sufficient conditions for the validity of a conjecture of Ewens (1969) for the survival probability of a slightly advantageous mutant gene.Research supported in part by NSF grants DMS-8803639 and DMS-9007182     

Viscosity of heparin as a function of dielectric constant and of desulfation

The effect of dielectric constant (D) of the solvent on the viscosity of heparin was examined using the relation \documentclass{article}\pagestyle{empty}\begin{document}$ \eta _{{\rm sp}} /c = [\eta ]_\infty (1 + k/\sqrt c) $\end{document}, where [η]_∞ is the shielded intrinsic viscosity obtained by extrapolating \documentclass{article}\pagestyle{empty}\begin{document}$ \eta _{{\rm sp}} /c\,{\rm vs}{\rm . }\,1/\sqrt c ) $\end{document} to infinite concentration, and k is an interaction parameter independent of the dielectric constant of the solvent. This equation was previously reported by the authors⁹ for describing the reduced viscosities of strong polyelectrolytes in salt-free polar solvents. It was found that the [η]_∞ of heparin increases linearly with increasing dielectric constant of the solvent whereas the k values were, within experimental error, independent of D in the range 54.7 < D < 93.2 examined. Graded hydrolysis of heparin from its acid form (heparinic acid) at 57°C resulted in samples of varying degree of desulfation with corresponding decrease in biological activity. It was found that both [η]_∞ and k decrease with increasing desulfation.     

Simulation of glucose isomerase reactor: optimum operating temperature

A design equation for immobilized glucose isomerase (IGI) packed bed reactor is developed assuming enzyme deactivation and substrate protection. The developed equation is used to simulate the performance of the reactor at various temperatures (50–80 °C). Enzyme deactivation is significant at high temperature. Substrate protection showed to have significant effect in reducing enzyme deactivation and increasing the enzyme half-life. Factors affecting the optimum operating temperature are discussed. The optimum operating temperature is greatly influenced by the operating period and to a lesser extent with both initial glucose concentration and glucose conversion.Two modes of reactor operation are tested i.e., constant feed flow rate and constant conversion. Reactor operating at constant conversion is more productive than reactor operating at constant flow rate if the working temperature is higher than the optimum temperature. Although at lower temperatures than the optimum, the two modes of operation give the same result.List of Symbols a residual enzyme activity - E [mg/l] concentration of active enzyme - E _a [kJ/mole] activation energy - E ₀ [mg/l] initial concentration of active enzyme - k [Specific] kinetic parameter - k _d [h^–1] first order thermal deactivation rate constant - k _e equilibrium constant - k _m [mole/l] apparent Michaelis constant - k _p [mole/l] Michaelis constant for product - k _s [mole/l] Michaelis constant for substrate - k ₀ [Specific] pre-exponential factor - Q [1/h] volumetric flow rate - ¯Q [1/h] average volumetric flow rate - R [kJ/mol·k] ideal gas constant - s [mole/l] apparent substrate concentration - s [mole/l] substrate concentration - s _e [mole/l] substrate concentration at equilibrium - s ₀ [mole/l] substrate concentration at reactor inlet - p [mole/l] product concentration - p _e [mole/l] product concentration at equilibrium - P _r [mole fructose/l·h] reactor productivity - T [k] temperature - t [h] time - t _p [h] operating time - V [l] reactor volume - v [mole/l·h] reaction rate - v [mole/l] reaction rate under enzyme deactivation and substrate protection - v _m [mole/l·h] maximum apparent reaction rate - v _p [mole/l·h] maximum reaction rate for product - v _s [mole/l·h] maximum reaction rate for substrate - x substrate fractional conversion - x _e substrate fractional conversion at equilibrium Greek Symbols effectiveness factor - mean effectiveness factor - substrate protection factor - [h] residence time - [h] average residence time - ₀ [h] initial residence time     

Light-controlled inhibition of hypocotyl growth inSinapis alba L. seedlings

Fluence rate-response curves were determined for the inhibition of hypocotyl growth in 54 h old dark-grownSinapis alba L. seedlings by continuous or hourly 5 min red light irradiation (24 h). In both cases a fluence rate-dependence was observed. More than 90% of the continuous light effect could be substituted for by hourly light pulses if the total fluence of the two different light regimes was the same. Measurements of the far red absorbing form of phytochrome ([P _fr]) and [P _fr]/[P tot] (total phytochrome) showed a strong fluence rate-dependence under continuous and pulsed light which partially paralleled the fluence rate-response curves for the inhibition of the hypocotyl growth.Abbreviations R red - HIR high irradiance response - P _rfr phytochrome in its red, far-red absorbing form - [P _tot]=[P _r]+[P _fr] =k ₁/(k ₁+k ₂): photoequilibrium of phytochrome at wavelength , wherebyk _1,2 rate constants ofP _rP _fr,P _frP _r photoconversion - [P _fr]/[P _tot]     

Distance matrix-based approach to protein structure prediction

Much structural information is encoded in the internal distances; a distance matrix-based approach can be used to predict protein structure and dynamics, and for structural refinement. Our approach is based on the square distance matrix D = [r _ij ² ] containing all square distances between residues in proteins. This distance matrix contains more information than the contact matrix C, that has elements of either 0 or 1 depending on whether the distance r _ij is greater or less than a cutoff value r _cutoff. We have performed spectral decomposition of the distance matrices $ {\mathbf{D}} = \sum {\lambda_{k} {\mathbf{v}}_{k} {\mathbf{v}}_{k}^{T} } Much structural information is encoded in the internal distances; a distance matrix-based approach can be used to predict protein structure and dynamics, and for structural refinement. Our approach is based on the square distance matrix D = [r _ij²] containing all square distances between residues in proteins. This distance matrix contains more information than the contact matrix C, that has elements of either 0 or 1 depending on whether the distance r _ij is greater or less than a cutoff value r _cutoff. We have performed spectral decomposition of the distance matrices , in terms of eigenvalues and the corresponding eigenvectors and found that it contains at most five nonzero terms. A dominant eigenvector is proportional to r ²—the square distance of points from the center of mass, with the next three being the principal components of the system of points. By predicting r ² from the sequence we can approximate a distance matrix of a protein with an expected RMSD value of about 7.3 ?, and by combining it with the prediction of the first principal component we can improve this approximation to 4.0 ?. We can also explain the role of hydrophobic interactions for the protein structure, because r is highly correlated with the hydrophobic profile of the sequence. Moreover, r is highly correlated with several sequence profiles which are useful in protein structure prediction, such as contact number, the residue-wise contact order (RWCO) or mean square fluctuations (i.e. crystallographic temperature factors). We have also shown that the next three components are related to spatial directionality of the secondary structure elements, and they may be also predicted from the sequence, improving overall structure prediction. We have also shown that the large number of available HIV-1 protease structures provides a remarkable sampling of conformations, which can be viewed as direct structural information about the dynamics. After structure matching, we apply principal component analysis (PCA) to obtain the important apparent motions for both bound and unbound structures. There are significant similarities between the first few key motions and the first few low-frequency normal modes calculated from a static representative structure with an elastic network model (ENM) that is based on the contact matrix C (related to D), strongly suggesting that the variations among the observed structures and the corresponding conformational changes are facilitated by the low-frequency, global motions intrinsic to the structure. Similarities are also found when the approach is applied to an NMR ensemble, as well as to atomic molecular dynamics (MD) trajectories. Thus, a sufficiently large number of experimental structures can directly provide important information about protein dynamics, but ENM can also provide a similar sampling of conformations. Finally, we use distance constraints from databases of known protein structures for structure refinement. We use the distributions of distances of various types in known protein structures to obtain the most probable ranges or the mean-force potentials for the distances. We then impose these constraints on structures to be refined or include the mean-force potentials directly in the energy minimization so that more plausible structural models can be built. This approach has been successfully used by us in 2006 in the CASPR structure refinement ().     

Hybrid maize breeding with doubled haploids: I. One-stage versus two-stage selection for testcross performance

Optimum allocation of resources is of fundamental importance for the efficiency of breeding programs. The objectives of our study were to (1) determine the optimum allocation for the number of lines and test locations in hybrid maize breeding with doubled haploids (DHs) regarding two optimization criteria, the selection gain ΔG _k and the probability P _k of identifying superior genotypes, (2) compare both optimization criteria including their standard deviations (SDs), and (3) investigate the influence of production costs of DHs on the optimum allocation. For different budgets, number of finally selected lines, ratios of variance components, and production costs of DHs, the optimum allocation of test resources under one- and two-stage selection for testcross performance with a given tester was determined by using Monte Carlo simulations. In one-stage selection, lines are tested in field trials in a single year. In two-stage selection, optimum allocation of resources involves evaluation of (1) a large number of lines in a small number of test locations in the first year and (2) a small number of the selected superior lines in a large number of test locations in the second year, thereby maximizing both optimization criteria. Furthermore, to have a realistic chance of identifying a superior genotype, the probability P _k of identifying superior genotypes should be greater than 75%. For budgets between 200 and 5,000 field plot equivalents, P _k > 75% was reached only for genotypes belonging to the best 5% of the population. As the optimum allocation for P _k (5%) was similar to that for ΔG _k , the choice of the optimization criterion was not crucial. The production costs of DHs had only a minor effect on the optimum number of locations and on values of the optimization criteria. C. Friedrich H. Longin and H. Friedrich Utz contributed equally to this work.