Allosteric control model of bone remodelling containing periodical modes

To help to understand the modelling process that occurs when a scaffold is implanted it is vital to understand the rather complex bone remodelling process prevalent in native bone. We have formulated a mathematical model that predicts osteoactivity both in scaffolds, as well as in bone in vivo and could set a basis for the more detailed allosteric models. The model is extended towards a bio-cybernetic vision of basic multicellular unit (BMU) action, when some of the regulation loops have been modified to reflect the allosteric control mechanisms, developed by Michaels-Menten, Hill, Koshland-Nemethy-Filmer, Monod-Wyman-Changeux. By implementation of this approach a four-dimensional system was obtained that shows steady cyclic behaviour using a wide range of constants with clear biological meaning. We have observed that a local steady state appears as a limiting cycle in multi-dimensional phase space and this is discussed in this paper. Physiological interpretation of this limiting four-dimension cycle possibly related to a conservative-like value has been proposed. Analysis and simulation of the model has shown an analogy between this conservative value, as a kind of substrate-energy regenerative potential of the bone remodelling system with a molecular nature, and to the classical physical value--energy. This dynamic recovery potential is directed against both mechanical and biomechanical damage to the bone. Furthermore, the current model has credibility when compared to the normal bone remodelling process. In the framework of widely recognised Hill mechanisms of allosteric regulation the cyclic attractor, described formerly for a pure cellular model, prevails for different forms of feedback control. This result indicates the viability of the proposed existence of a conservative value (analogous to energy) that characterises the recovery potential of the bone remodelling cycle. Linear stability analysis has been performed in order to determine the robustness of the basic state, however, additional work is required to study a wider range of constants.     

Phenomenological model of bone remodeling cycle containing osteocyte regulation loop

Biological parameters, such as bone resorption and formation constants, are important variables to achieve optimised hard tissue scaffolds design. To help to understand the modelling process that occurs when a scaffold is implanted it is vital to understand the rather complex bone remodeling process prevalent in native bone. One approach to developing a mathematical model that predicts osteoactivity both in scaffolds, as well as in bone in vivo, is based on a bio-cybernetic vision of basic multicellular unit (BMU) action -. In the case of the model presented in this paper, an additional loop of regulation based on osteocyte activity has been added. This approach has resulted in a four-dimensional system, which shows steady-quasi-cyclic behaviour using a particular range of constants with real biological meaning. The initial findings suggesting that the basic steady-state appears as a torus in multidimensional phase space have been discussed. The existence of this surface in the osteoclasts-osteoblasts-osteocytes-bone subspace indicates that there is a first integral for this dynamic system. Biological and physical interpretation of this integral as a conservative value has been proposed. It is possible to draw an analogy between this conservative value, as a kind of substrate-energy regenerative potential of the bone remodeling system with a molecular nature, to the classical physical value (energy). There are clear indications that there is recovering potential within the BMU that results in a steady operating genetically predominated bone remodeling process. This recovering potential is directed against both mechanical and biomechanical damage to the bone. The current model has credibility when compared to the normal bone remodeling process. However, additional work is required to study a wider range of constants.     

Bone remodelling algorithms incorporating both strain and microdamage stimuli

Biomechanical theories to predict bone remodelling have used either mechanical strain or microdamage as the stimulus driving cellular responses. Even though experimental data have implicated both stimuli in bone cell regulation, a mechano-regulatory system incorporating both stimuli has not yet been proposed. In this paper, we test the hypothesis that bone remodelling may be regulated by signals due to both strain and microdamage. Four mechano-regulation algorithms are studied where the stimulus is: strain, damage, combined strain/damage, and either strain or damage with damage-adaptive remodelling prioritised when damage is above a critical level. Each algorithm is implemented with both bone lining cell (surface) sensors and osteocyte cell (internal) sensors. Each algorithm is applied to prediction of a bone multicellular unit (BMU) remodelling on the surface of a bone trabecula. It is predicted that a regulatory system capable of responding to changes in either strain or microdamage but which prioritises removal of damaged bone when damage is above a critical level, is the only one that provides a plausible prediction of BMU behaviour. A mechanism for this may be that, below a certain damage threshold, osteocyte processes can sense changes in strain and fluid flow but above the threshold damage interferes with the signalling mechanism, or causes osteocyte apoptosis so that a remodelling response occurs to remove the dead osteocytes.     

A novel mathematical model of bone remodelling cycles for trabecular bone at the cellular level

After an initial phase of growth and development, bone undergoes a continuous cycle of repair, renewal and optimisation by a process called remodelling. This paper describes a novel mathematical model of the trabecular bone remodelling cycle. It is essentially formulated to simulate a remodelling event at a fixed position in the bone, integrating bone removal by osteoclasts and formation by osteoblasts. The model is developed to construct the variation in bone thickness at a particular point during the remodelling event, derived from standard bone histomorphometric analyses. The novelties of the approach are the adoption of a predator-prey model to describe the dynamic interaction between osteoclasts and osteoblasts, using a genetic algorithm-based solution; quantitative reconstruction of the bone remodelling cycle; and the introduction of a feedback mechanism in the bone formation activity to co-regulate bone thickness. The application of the model is first demonstrated by using experimental data recorded for normal (healthy) bone remodelling to predict the temporal variation in the number of osteoblasts and osteoclasts. The simulated histomorphometric data and remodelling cycle characteristics compare well with the specified input data. Sensitivity studies then reveal how variations in the model's parameters affect its output; it is hoped that these parameters can be linked to specific biochemical factors in the future. Two sample pathological conditions, hypothyroidism and primary hyperparathyroidism, are examined to demonstrate how the model could be applied more broadly, and, for the first time, the osteoblast and osteoclast populations are predicted for these conditions. Further data are required to fully validate the model's predictive capacity, but this work shows it has potential, especially in the modelling of pathological conditions and the optimisation of the treatment of those conditions.     

Bone remodelling analysis of a bovine femur for a veterinary implant design

The response of bovine bone to the presence of an implant is analysed with the aim of simulating bone remodelling in a developing model of a polymeric intramedullary interlocking nail for veterinary use. A 3-D finite element model of the femur diaphysis is built based on computed tomography images and using a CAD-based modelling pipeline. The bone remodelling process after the surgery is analysed and compared with the healthy bone. The remodelling law assumes that bone adapts to the mechanical environment. For the analyses a consistent set of loads is determined for the bovine walk cycle. The remodelling results reproduce the morphologic features of bone and provide evidence of the difference on the bone behaviour when comparing metallic and polymeric nails. Our findings indicate that an intramedullary polymeric nail has the advantage over the metallic one of improving long-term bone healing and possibly avoiding the need of the implant removal.     

An agent based model for real-time signaling induced in osteocytic networks by mechanical stimuli

We recently observed that insertion of unloaded rest between each load cycle substantially enhanced bone formation induced by mild loading regimens. To begin to explore this result, we have developed an agent based model for real-time signaling induced when osteocytic networks are challenged by mechanical stimuli. In the model, activity induced in individual osteocytes were governed by the following cellular functions: (1) threshold levels of tissue strain magnitudes were required to initiate and maximally activate cells, (2) cell activity beyond thresholds were propagated within localized neighborhoods and influenced recipient cell activity, (3) cellular activity was modulated by 'molecular' stores and the rates at which stores were replenished when cells were quiescent. Using this model, the real-time response of osteocyte networks was determined as the average of individual cell activity. While not explicitly embedded within the model, interactions between cellular functions served as positive, negative, and end-point feedback mechanisms and resulted in unique real-time network responses to distinct mechanical stimuli. Specifically, the real-time network response to cyclic stimuli consisted of a large magnitude transient followed by low-level steady state fluctuations, while rest-inserted stimuli induced multiple secondary transients. Analysis of interaction patterns suggested that rest-inserted stimuli induced this enhanced and sustained signaling within osteocytic networks by enabling cell recovery of expended molecular stores and by efficiently utilizing properties inherent to cell-cell communication in bone. Importantly, this emergence based approach suggested mechanisms potentially underlying the benefit of rest-inserted stimuli and provides a unique framework for a broader exploration of mechanotransduction function within bone.     

Emerging maps of allosteric regulation in cellular networks

Allosteric regulation is classically defined as action at a distance, where a perturbation outside of a protein active site affects function. While this definition has motivated many studies of allosteric mechanisms at the level of protein structure, translating these insights to the allosteric regulation of entire cellular processes – and their crosstalk – has received less attention, despite the broad importance of allostery for cellular regulation foreseen by Jacob and Monod. Here, we revisit an evolutionary model for the widespread emergence of allosteric regulation in colocalized proteins, describe supporting evidence, and discuss emerging advances in mapping allostery in cellular networks that link precise and often allosteric perturbations at the molecular level to functional changes at the pathway and systems levels.     

A cell based modelling framework for skeletal tissue engineering applications

In this study, a cell based lattice free modelling framework is proposed to study cell aggregate behaviour in bone tissue engineering applications. The model encompasses cell-to-cell and cell–environment interactions such as adhesion, repulsion and drag forces. Oxygen, nutrients, waste products, growth factors and inhibitors are explicitly represented in the model influencing cellular behaviour. Furthermore, a model for cell metabolism is incorporated representing the basic enzymic reactions of glycolysis and the Krebs cycle. Various types of cell death such as necrosis, apoptosis and anoikis are implemented. Finally, an explicit model of the cell cycle controls the proliferation process, taking into account the presence or absence of various metabolites, sufficient space and mechanical stress. Several examples are presented demonstrating the potential of the modelling framework. The behaviour of a synchronised cell aggregate under ideal circumstances is simulated, clearly showing the different stages of the cell cycle and the resulting growth of the aggregate. Also the difference in aggregate development under ideal (normoxic) and hypoxic conditions is simulated, showing hypoxia induced necrosis mainly in the centre of the aggregate grown under hypoxic conditions. The next step in this research will be the application of this modelling framework to specific experimental set-ups for bone tissue engineering applications.     

A theoretical study of bone remodelling under PEMF at cellular level

Pulsed electromagnetic field (PEMF) devices have been used clinically to slow down osteoporosis and accelerate the healing of bone fractures for many years. However, the underlying mechanism by which bone remodelling under PEMF is regulated remains poorly understood. In this paper, a mathematical model of bone cell population of bone remodelling under PEMF at cellular level is developed to address this issue for the first time. On the basis of this model and control theory, parametric study of control mechanisms is carried out and a number of possible control mechanisms are identified. These findings will help further the understanding of bone remodelling under PEMF and advance therapies and pharmacological developments in clinical trials.     

A theoretical study of bone remodelling under PEMF at cellular level

Pulsed electromagnetic field (PEMF) devices have been used clinically to slow down osteoporosis and accelerate the healing of bone fractures for many years. However, the underlying mechanism by which bone remodelling under PEMF is regulated remains poorly understood. In this paper, a mathematical model of bone cell population of bone remodelling under PEMF at cellular level is developed to address this issue for the first time. On the basis of this model and control theory, parametric study of control mechanisms is carried out and a number of possible control mechanisms are identified. These findings will help further the understanding of bone remodelling under PEMF and advance therapies and pharmacological developments in clinical trials.     

Comparative analysis of bone remodelling models with respect to computerised tomography-based finite element models of bone

Subject-specific finite element models are an extensively used tool for the numerical analysis of the biomechanical behaviour of human bones. However, bone modelling is not an easy task due to the complex behaviour of bone tissue, involving non-homogeneous and anisotropic mechanical properties. Moreover, bone is a living tissue and therefore its microstructure and mechanical properties evolve with time in a known process called bone remodelling. This phenomenon has been widely studied, many being the numerical models that have been formulated to predict density distribution and its evolution in several bones. The aim of the present study is to assess the capability of a bone remodelling model to predict the bone density distribution of different types of human bone (femur, tibia and mandible) comparing the obtained results with the bone density estimated by means of computerised tomography. Good accuracy was observed for the bone remodelling predictions including the thickness of the cortical layer.     

Mathematical model of carbohydrate energy metabolism. Interaction between glycolysis, the Krebs cycle and the H-transporting shuttles at varying ATPase load

A simple mathematical model for carbohydrate energy metabolism based on the stoichiometic structure of glycolysis, the Krebs cycle and oxidative phosphorylation is proposed. The only allosteric regulation involved in the model is phosphofructokinase activation by AMP. Simple as it is, the model can explain the following properties of carbohydrate metabolism: a drastic rise of the rate of glucose consumption during transition to a higher level of ATPase load; stabilization of ATP and an increase of the steady state rates of glycolysis and oxidation of cytoplasmic NADH by the H-transporting shuttles and of pyruvate in the Krebs cycle with increasing rate of the ATPase load; activation of glycolysis and a decrease of the rate of oxidative phosphorylation following an inhibition of the H-transporting shuttles. The mechanisms of the coordinated changes in the steady state rates of glycolysis, the H-transporting shuttles and the Krebs cycle at varying ATPase load in the cell are discussed.     

Oscillations in calcium-cyclic AMP control loops form the basis of pacemaker activity and other high frequency biological rhythms.

In this paper theoretical and experimental evidence is presented which indicates that oscillations in internal calcium and cyclic AMP concentrations due to an instability in their common control loops are possible and indeed may be widespread. Further, it is demonstrated that fluctuations in various cellular properties, in particular membrane potential, are a direct consequence of these second messenger oscillations. Given the central importance of calcium and cyclic AMP to the regulation of metabolism, these oscillations would influence most metabolic processes especially rhythmic behaviour. We propose that these oscillations form the basis of several biological rhythms including, potential oscillations in cardiac pacemaker cells, neurones and insulin secreting β-cells, the minute rhythm in smooth muscle, cyclic AMP pulses in Dictyostelium, rhythmical cytoplasmic streaming in Physarum and transepitheliel potential oscillations in Calliphora salivary gland. This model makes possible an explanation of the frequency and amplitude effects of hormones.     

Improving the damage accumulation in a biomechanical bone remodelling model

We extend, reformulate and analyse a phenomenological model for bone remodelling. The original macrobiomechanical model (MBM), proposed by Hazelwood et al. [J Biomech 2001; 34:299–308], couples a population equation for the cellular activities of the basic multicellular units (BMUs) in the bone and a rate equation to account for microdamage and repair. We propose to account for bone failure under severe overstressing by incorporating a Paris-like power-law damage accumulation term. The extended model agrees with the Hazelwood et al. predictions when the bone is under-stressed, and allows for suitably loaded bones to fail, in agreement with other MBM and experimental data regarding damage by fatigue. We numerically solve the extended model using a convergent algorithm and show that for unchanging loads, the stationary solution captures fully the model behaviour. We compute and analyse the stationary solutions. Our analysis helps guide additional extensions to this and other BMU activity based models.     

Steroid hormone antagonism and a cyclic model of receptor kinetics

Steady state agonist-antagonist relations have been derived for a general version of a cyclic model of glucocorticoid-receptor kinetics. The model was previously shown to account quantitatively for the transient and steady state distribution of hormone-receptor complexes formed in thymus cells by several glucocorticoids. Agonist-antagonist properties of a steroid in the model are expressed quantitatively by its "agonist activity" A, the steady state ratio of nuclear-bound to total complexes it forms. For a pure agonist A = 1, for a pure antagonist A = 0. This ratio is found to be independent of steroid concentration and a function only of the rate constants of reactions involving complexes formed by the steroid. Analysis of the dependence of A on each rate constant reveals how each reaction in the cyclic model--activation, nuclear binding, dissociation of activated and nuclear-bound complexes--influences antagonist properties. The steady state interaction of an antagonist with an agonist is shown to be governed by relations that are indistinguishable from competition relations for the simplest equilibrium system, and to yield dose-response curves that are very similar to those produced by two-state allosteric models of steroid hormone antagonism, despite the fact that the cyclic model includes no allosteric mechanisms. With steroids for which relevant rate constants can be measured, the model is directly testable. Limitations of the model arise from lack of information about the nuclear events that lead to biological activity following binding of activated complexes to the nucleus.     

Application of an anisotropic bone-remodelling model based on a damage-repair theory to the analysis of the proximal femur before and after total hip replacement

In this work, a new model for internal anisotropic bone remodelling is applied to the study of the remodelling behaviour of the proximal femur before and after total hip replacement (THR). This model considers bone remodelling under the scope of a general damage-repair theory following the principles of continuum damage mechanics. A "damage-repair" tensor is defined in terms of the apparent density and Cowin's "fabric tensor", respectively, associated with porosity and directionality of the trabeculae. The different elements of a thermodynamically consistent damage theory are established, including resorption and apposition criteria, evolution law and rate of remodelling. All of these elements were introduced and discussed in detail in a previous paper (García, J. M., Martinez, M. A., Doblaré, M., 2001. An anisotrophic internal-external bone adaptation model based on a combination of CAO and continuum damage mechanics technologies. Computer Methods in Biomechanics and Biomedical Engineering 4(4), 355-378.), including the definition of the proposed mechanical stimulus and the qualitative properties of the model. In this paper, the fundamentals of the proposed model are briefly reviewed and the computational aspects of its implementation are discussed. This model is then applied to the analysis of the remodelling behaviour of the intact femur obtaining densities and mass principal values and directions very close to the experimental data. The second application involved the proximal femoral extremity after THR and the inclusion of an Exeter prosthesis. As a result of the simulation process, some well-known features previously detected in medical clinics were recovered, such as the stress yielding effect in the proximal part of the implant or the enlargement of the cortical layer at the distal part of the implant. With respect to the anisotropic properties, bone microstructure and local stiffness are known to tend to align with the stress principal directions. This experimental fact is mathematically proved in the framework of this remodelling model and clearly shown in the results corresponding to the intact femur. After THR the degree of anisotropy decreases tending, specifically in the proximal femur, to a more isotropic behaviour.     

Voltage sensor movement and cAMP binding allosterically regulate an inherently voltage-independent closed-open transition in HCN channels

The hyperpolarization-activated cyclic nucleotide-modulated cation (HCN) channels are regulated by both membrane voltage and the binding of cyclic nucleotides to a cytoplasmic, C-terminal cyclic nucleotide-binding domain (CNBD). Here we have addressed the mechanism of this dual regulation for HCN2 channels, which activate with slow kinetics that are strongly accelerated by cAMP, and HCN1 channels, which activate with rapid kinetics that are weakly enhanced by cAMP. Surprisingly, we find that the rate of opening of HCN2 approaches a maximal value with extreme hyperpolarization, indicating the presence of a voltage-independent kinetic step in the opening process that becomes rate limiting at very negative potentials. cAMP binding enhances the rate of this voltage-independent opening step. In contrast, the rate of opening of HCN1 is much greater than that of HCN2 and does not saturate with increasing hyperpolarization over the voltage range examined. Domain-swapping chimeras between HCN1 and HCN2 reveal that the S4-S6 transmembrane region largely determines the limiting rate in opening kinetics at negative voltages. Measurements of HCN2 tail current kinetics also reveal a voltage-independent closing step that becomes rate limiting at positive voltages; the rate of this closing step is decreased by cAMP. These results are consistent with a cyclic allosteric model in which a closed-open transition that is inherently voltage independent is subject to dual allosteric regulation by voltage sensor movement and cAMP binding. This mechanism accounts for several properties of HCN channel gating and has potentially important physiological implications.     

Targeting ubiquitin specific protease 7 in cancer: A deubiquitinase with great prospects

Deubiquitinase (DUB)‐mediated cleavage of ubiquitin chain balances ubiquitination and deubiquitination for determining protein fate. USP7 is one of the best characterized DUBs and functionally important. Numerous proteins have been identified as potential substrates and binding partners of USP7; those play crucial roles in diverse array of cellular and biological processes including tumour suppression, cell cycle, DNA repair, chromatin remodelling, and epigenetic regulation. This review aims at summarizing the current knowledge of this wide association of USP7 with many cellular processes that enlightens the possibility of abnormal USP7 activity in promoting oncogenesis and the importance of identification of specific inhibitors.     

Bone remodelling and tumour grade modifications induced by interactions between bone and swarm rat chondrosarcoma

Chondrosarcoma is currently defined as a malignant cartilage tumour arising de novo or within a pre-existing benign cartilage tumour. Chondrosarcoma can be surgically resected, but all grades have significant rates of local recurrence. The purpose of the present study was to develop an animal intraosseous chondrosarcoma model simulating the progression of human chondrosarcoma and elucidating its behaviour and biology. An intraosseous Swarm rat model was designed to assess interactions between bone and chondrosarcoma. A comparison of tumour grading was carried out according to transplantation site. The effects of chondrosarcoma cells (SRC cells) on the mineralisation capacities of osteoblasts and on osteoclast differentiation were studied in relation to modifications observed in vivo at the cellular level. Transplantation of Swarm rat chondrosarcoma within bone marrow or contiguous to induced periosteal lesions led to extensive bone remodelling with trabecular bone rarefaction and periosteal apposition. Transplantation in close contact to bone but without any periosteal lesion had no effect on bone, suggesting that bone healing factors interact with tumour development. With the intramedullary model, the development of tumours of different grade confirms that bone environment is an important factor in malignancy. A decrease of bone nodule formation was noted after cocultures of SRC cells with rat bone marrow, but there was no modification of osteoclast differentiation after cultures of total rabbit bone cells with SRC cells. These data reveal the importance of interactions between bone environment and tumour in inducing bone remodelling and variations in tumour malignancy.