Propagation of external regulation and asynchronous dynamics in random Boolean networks.

Boolean networks and their dynamics are of great interest as abstract modeling schemes in various disciplines, ranging from biology to computer science. Whereas parallel update schemes have been studied extensively in past years, the level of understanding of asynchronous updates schemes is still very poor. In this paper we study the propagation of external information given by regulatory input variables into a random Boolean network. We compute both analytically and numerically the time evolution and the asymptotic behavior of this propagation of external regulation (PER). In particular, this allows us to identify variables that are completely determined by this external information. All those variables in the network that are not directly fixed by PER form a core which contains, in particular, all nontrivial feedback loops. We design a message-passing approach allowing to characterize the statistical properties of these cores in dependence of the Boolean network and the external condition. At the end we establish a link between PER dynamics and the full random asynchronous dynamics of a Boolean network.

Core percolation and onset of complexity in boolean networks.

The determination and classification of fixed points of large Boolean networks is addressed in terms of a constraint-satisfaction problem. We develop a general simplification scheme that, removing all those variables and functions belonging to trivial logical cascades, returns the computational core of the network. The transition line from an easy to a complex regulatory phase is described as a function of the parameters of the model, identifying thereby both theoretically and algorithmically the relevant regulatory variables.

Polynomial iterative algorithms for coloring and analyzing random graphs.

We study the graph coloring problem over random graphs of finite average connectivity c. Given a number q of available colors, we find that graphs with low connectivity admit almost always a proper coloring whereas graphs with high connectivity are uncolorable. Depending on q, we find with a one-step replica-symmetry breaking approximation the precise value of the critical average connectivity c(q). Moreover, we show that below c(q) there exists a clustering phase c in [c(d),c(q)] in which ground states spontaneously divide into an exponential number of clusters. Furthermore, we extended our considerations to the case of single instances showing consistent results. This leads us to propose a different algorithm that is able to color in polynomial time random graphs in the hard but colorable region, i.e., when c in [c(d),c(q)].

Coloring random graphs.

We study the graph coloring problem over random graphs of finite average connectivity c. Given a number q of available colors, we find that graphs with low connectivity admit almost always a proper coloring, whereas graphs with high connectivity are uncolorable. Depending on q, we find the precise value of the critical average connectivity c(q). Moreover, we show that below c(q) there exists a clustering phase c in [c(d),c(q)] in which ground states spontaneously divide into an exponential number of clusters and where the proliferation of metastable states is responsible for the onset of complexity in local search algorithms.

Hiding solutions in random satisfiability problems: a statistical mechanics approach.

A major problem in evaluating stochastic local search algorithms for NP-complete problems is the need for a systematic generation of hard test instances having previously known properties of the optimal solutions. On the basis of statistical mechanics results, we propose random generators of hard and satisfiable instances for the 3-satisfiability problem. The design of the hardest problem instances is based on the existence of a first order ferromagnetic phase transition and the glassy nature of excited states. The analytical predictions are corroborated by numerical results obtained from complete as well as stochastic local algorithms.

Minimal vertex covers on finite-connectivity random graphs: a hard-sphere lattice-gas picture.

The minimal vertex-cover (or maximal independent-set) problem is studied on random graphs of finite connectivity. Analytical results are obtained by a mapping to a lattice gas of hard spheres of (chemical) radius 1, and they are found to be in excellent agreement with numerical simulations. We give a detailed description of the replica-symmetric phase, including the size and entropy of the minimal vertex covers, and the structure of the unfrozen component which is found to percolate at a connectivity c approximately 1.43. The replica-symmetric solution breaks down at c=e approximately 2.72. We give a simple one-step replica-symmetry-broken solution, and discuss the problems in the interpretation and generalization of this solution.

Simplest random K-satisfiability problem.

We study a simple and exactly solvable model for the generation of random satisfiability problems. These consist of gammaN random boolean constraints which are to be satisfied simultaneously by N logical variables. In statistical-mechanics language, the considered model can be seen as a diluted p-spin model at zero temperature. While such problems become extraordinarily hard to solve by local search methods in a large region of the parameter space, still at least one solution may be superimposed by construction. The statistical properties of the model can be studied exactly by the replica method and each single instance can be analyzed in polynomial time by a simple global solution method. The geometrical and topological structures responsible for dynamic and static phase transitions as well as for the onset of computational complexity in the local search method are thoroughly analyzed. Numerical analysis on very large samples allows for a precise characterization of the critical scaling behavior.

Typical solution time for a vertex-covering algorithm on finite-connectivity random graphs.

We analytically describe the typical solution time needed by a backtracking algorithm to solve the vertex-cover problem on finite-connectivity random graphs. We find two different transitions: The first one is algorithm dependent and marks the dynamical transition from linear to exponential solution times. The second one gives the maximum computational complexity, and is found exactly at the threshold where the system undergoes an algorithm-independent phase transition in its solvability. Analytical results are corroborated by numerical simulations.

Number of guards needed by a museum: a phase transition in vertex covering of random graphs.

In this Letter we study the NP-complete vertex cover problem on finite connectivity random graphs. When the allowed size of the cover set is decreased, a discontinuous transition in solvability and typical-case complexity occurs. This transition is characterized by means of exact numerical simulations as well as by analytical replica calculations. The replica symmetric phase diagram is in excellent agreement with numerical findings up to average connectivity e, where replica symmetry becomes locally unstable.

Activation of luminal Na+/H+ exchange in distal nephron of frog kidney. An early response to aldosterone.

Increased chronic intake of K+ induces H+ and K+ secretion in amphibian distal tubule, paralleled by an elevation of plasma aldosterone. The present experiments test whether the mineralocorticoid hormone is responsible for the alteration of ion transport. The blood capillaries of the isolated kidneys of NaCl-adapted (i.e. aldosterone-suppressed) Rana pipiens were perfused with HEPES-buffered amphibian Ringer solution (pH 7.8). Limiting intraluminal pH (pHlu) was measured continuously with pH-sensitive microelectrodes while aldosterone (3 X 10(-7) to 3 X 10(-6) mol/l) was applied in the peritubular perfusate. Concomitant with a decrease of the lumen-positive transepithelial potential (Vte) from 8.5 +/- 1.1 mV to 4.0 +/- 0.6 mV pHlu dropped from 7.73 +/- 0.02 to a new steady-state value of 7.17 +/- 0.05 within 60 to 180 min of aldosterone administration. Significant luminal acidification occurred already 20 min after application of aldosterone. Luminal addition of 10(-3) mol/l amiloride reversed luminal acidification to a pHlu of 7.68 +/- 0.04; at the same time Vte recovered partially. Pretreatment of the distal tubules with spironolactone prevented the aldosterone-induced acidification of the tubule fluid. We conclude that in early distal tubule of the amphibian kidney aldosterone--after interaction with cytoplasmic receptors--activates the luminal, amiloride-inhibitable Na+/H+ exchanger. This mechanism could explain enhanced H+ secretion found in the K+ adapted animal.

Aldosterone activates Na+/H+ exchange and raises cytoplasmic pH in target cells of the amphibian kidney.

The hypothesis was tested if the mineralocorticoid hormone aldosterone stimulates Na+/H+ exchange in "giant cells" fused from individual target cells of the distal nephron of the frog kidney. By means of microelectrodes, steady-state intracellular pH (pHi) and pHi recovery from an acid load were recorded continuously while the fused cells were exposed to aldosterone. Twenty minutes after addition of the hormone, pHi started to rise and reached a new steady state after about 60 min (delta pHi = 0.28 +/- 0.01). After hormone treatment, pHi recovered significantly faster in response to an intracellular acid load. The diuretic drug amiloride blocked pHi recovery. Experiments in intact tubules showed that aldosterone induces H+ and K+ secretion. Thus, intracellular alkalosis, mediated by Na+/H+ exchange, could serve as a signal that activates pH-sensitive K+ channels of the luminal cell membrane.

Relationship between luminal Na+/H+ exchange and luminal K+ conductance in diluting segment of frog kidney.

Experiments were performed in the isolated perfused kidney of K+ adapted Rana pipiens to investigate the relationship between luminal K+ conductance and H+ transport in cells of the diluting segment. Inhibition of luminal Na+/H+ exchange by amiloride or by omission of luminal Na+ blocked luminal K+ conductance. Acidification of the kidney perfusate by elevation of pCO2 also reduced luminal K+ conductance. This effect could be prevented by furosemide. Since the steepest transcellular Na+ potential difference, directed from the lumen into the cell, is found when luminal Na+/Cl-/K+ cotransport is inhibited by furosemide, we conclude that luminal Na+/H+ exchange is most efficient at these conditions and thus could attenuate intracellular acidification.