We also examined the velocity and shear stress profiles for flow over the permeable layer for the limiting cases of α → 0 and α → ∞. The model predictions were compared with the experimental data for specific porous media and good agreement was found.While researches have focused on drag reduction of various coated surfaces such as superhydrophobic structures and polymer brushes, the insights tso understand the fundamental physics of the laminar skin friction coefficient and the related drag reduction due to the formation of finite velocity at porous surfaces is still relatively unknown. Herein, we quantitatively investigated the flow over a porous medium by developing a framework to model flow of a Newtonian fluid in a channel where the lower surface was replaced by various porous media. We showed that the flow drag reduction induced by the presence of the porous media depends on the values of the permeability parameter α = L/(MK) In fact, experimental studies of blood flow in microvessels revealed differences from analytical predictions based on in vitro observations2. It was found that the main reason for these differences is the presence of a relatively thick (~1 μm) layer of macromolecules bound to the endothelial cells lining microvessel walls3. This layer, which is located on the luminal surface of vascular cells, is called the endothelial glycocalyx layer (EGL)  Fig. . While t7.In addition to preventing adhesive molecular interaction between proteins in the red cell membrane and their ligands in the plasmalemma membrane of the endothelial cell, the EGL provides a red blood cell “exclusion zone” or “gap” between the flowing red blood cells and the endothelial membrane as shown in Fig. 8 further found the so-called “pop out” phenomenon in which at velocities <20 μm/s the red cells entered the EGL and, when motion was arrested, the red cells crushed the glycocalyx and filled nearly the entire lumen of the capillary. At velocities >20 μm/s, the red cells appeared to glide above the EGL, and there was a narrow intervening fluid gap. To describe the latter motion, a lubrication theory for soft porous highly compressible materials with periodic fiber arrays was developed8. Later a lubrication theory for randomly oriented soft porous media was also established9. Using the proposed lubrication theory, it was shown that the corresponding generated lift forces could be four orders of magnitude greater than those predicted by classical lubrication theory11.Feng and WeinbaumThe concept of the almost frictionless movement of red blood cells through tiny capillaries and the impact of the EGL on the red blood cell motion inspired us to examine a totally different scenario: analysis of the laminar skin friction coefficient and drag reduction in micorchannels where the bottom solid surface is replaced with a layer of porous material with permeability K and porosity ε. While the topic of drag reduction has been the focus of recent research on surfaces coated with a broad array of structures such as superhydrophobic structures and polymer brushes, the insights to understand the fundamental physics of the laminar skin friction coefficient and the related drag reduction due to the formation of a slip velocity at the surface of a porous media boundary in the pressure-driven flow is still not fully understood. Herein, we focus on answering the questions including: 1) What is the relation between the characteristics of the porous media  and the geometry of the channel with the resulting drag on the porous walls? 2) Can replacing an impermeable wall with a permeable surface be used as a new passive technique for reducing friction in a pressure-driven duct flow? 3) How the characteristics of porous media and channel geometry can be correlated to the drag force on the wall?16 provide reviews of related research developments in the literature. Researchers have investigated flow over sediment beds17, coral reefs and submerged vegetation canopies18, crop canopies and forests19, endothelial glycocalyx of blood vessels20, flow over carbon nanotubes (CNTs)21, and polymer brushes23, and the finite-Reynolds flow in a channel bounded by one or two porous media modeled by cubic sphere arrays24. The drag reduction due to the motion of a viscous fluid, in laminar and turbulent flows, over CNTs, over patterned surfaces, and over various other structured surfaces has been examined experimentally and theoretically46. However, to the best of our knowledge, no one has ever attempted to characterize the laminar drag reduction in a channel in the presence of porous walls. In this study, we present analytical description of the laminar skin friction coefficient and the corresponding drag reduction of the flow at the fluid-porous interface inside a channel with the lower surface replaced with various porous layers with Darcy permeability K, and porosity ε. The conditions for which the porous materials can be used as a new passive technique for reducing drag in laminar flow regimes were also described.Coupled flows through and over porous layers have been covered in several papers. A great number of publicationsμ is the dynamic viscosity of the fluid, and μe is the effective viscosity of fluid in the porous layer. The effective viscosity in porous media is widely discussed in previous studies47. The conclusion of these earlier studies is that for high porosity material, μe → μ. The boundary conditions on the two walls of the channel, namely at 48 we require that the velocity and fluid shear stress to be continuous at the fluid-porous interface , the core velocity profile is parabolic except near the interface, where the velocity drops to very small values. In addition, for large values of α, when the permeability of the porous material becomes low, the velocity profile inside the porous layer shows the characteristic of plug flow, except near the free fluid-porous medium interface , , the porous-wall surface would not provide much drag reduction in the channel even for large values of δ. As the height ratio increases to 2 where the height of the free-flow and the thickness of the fiber layer are the same, the maximum drag reduction in the system occurs.For a more clear comparison, variations of drag reduction DR% versus α are plotted in Fig. The drag reduction, DR%, as a function of δ for various permeability parameters 3, respectively. A peristaltic pump (Simply Pump Inc.) was used to pump the fluid through the system. The pump had a maximum head of 20 ft. A flowmeter  was also utilized for measuring the flow rate through the system. A dampener  damped the effect of pulsatility of the pump during the experiments. A pressure gauge (Omega Inc.) with testing range 0~4.0 inH2O was used to measure the pressure drop in the channel. Interchangeable surfaces were placed on the bottom of the channel to adjust the height of the free fluid and to keep it constant during the test. In these experiments, the lower solid wall was shifted to accommodate for the permeable layer boundary conditions, while maintaining the same free flow channel height. The porous media used was manufactured by Mountain Mist that were made of random array of fibers with 5% silk and 95% polyester.In order to validate our analytical predictions presented in previous sections, a series of experiments was conducted and the pressure drops across the channel for a range of conditions were measured. The schematic of the experimental set-up for pressure drop measurements is shown in Fig. α. This was because as can be observed in Fig. In order to obtain additional experimental data for various values of d is the sample length over which the pressure drop occurs, and A is the cross-sectional area of the porous layer. The experimental measurements of Darcy permeability for the two porous media used in the experiments led to the values of 7.2 × 10−8 m2 and 6.2 × 10−8 m2. The corresponding porosities were measured to be 0.97 and 0.98.Before conducting the channel flow tests, we measured the Darcy permeability K, of the sample porous materials by performing the Darcy experiments. According to Darcy’s law46DR%theo, was evaluated by using the experimental permeability parameter α and height ratio δ in equations  In the theoretical model it was assumed that the porous medium is distributed uniformly and the porous-fluid interface was smooth and at a fixed level. In the experiments, however, the distribution of fibers might have introduced variations in the level of fluid-porous media interface and also inhomogeneity of porosity or permeability near the interface, in addition to some surface roughness, 2) The experimental setup had a cross-sectional aspect ratio around (5:1), but in the analytical model the channel was considered to be two-dimensional and the 3D effects were neglected, 3) Correlations were made with Reynolds number that was based on the flow in the non-porous section. While this maybe a reasonable assumption for low permeability porous layer, it will introduce errors when permeability is high, and 4) The accuracy of the experimental measurement devices such as the pressure gauge and flow meter would also impact the experimental data.Figure 11. Drag reduction in the presence of a porous boundary could significantly advance microfluidic-based devices in a wide range of industrial applications, in particular in biomedical fields. In this study, we developed an analytical framework for predicting laminar fluid flows and the corresponding drag reductions in ducts when the bottom solid surface of the channel is replaced with permeable layers. Particular attention was given to the case when the porosity of the porous layer is close to 1. Based on the presented results the following conclusions are drawn:1/2 and the height ratio δ = H/L.The presented results showed that the flow drag reduction was induced when the lower solid surface of a channel was replaced by a porous medium. The amount of drag reduction was shown to depend on the permeability parameter α = L/(MK)The limiting behavior of the flow over the porous boundary for both α → 0 and α → ∞ were evaluated. It was found that as α → ∞ the velocity and shear stress profiles asymptotically approaches to the limit of the pressure-driven flow in a channel with solid walls with height of 2 L. In the case when α → 0, the velocity and shear stress profiles becomes the same as those of a wider channel with a height of 2 L + H.The model correlates the resulting reduction in drag to the characteristics of the permeable boundary surface. That is, with the knowledge of the channel geometry, the fluid bulk velocity, and the porosity and permeability of the porous wall, one can evaluate the drag reduction in the system.The presented model reveals the relation between the physical properties of the permeable layer and the features of the fluid flows including drag reduction that can be used for designing relevant engineering systems. For example, knowing the values of the permeability parameter and the height ratio provides guidance for designing laboratory-scaled-model that are dynamically similar to their industrial scale prototypes.Laminar flows over permeable surfaces occur in various engineering applicationsIn addition, for verifying the accuracy of the developed analytical model, several experiments were conducted for large values of α with specific height ratios and permeability parameters. Comparison of the results showed good agreements between the experimental data and analytical predictions for drag reduction.Studying the effect of height variations of the fluid-porous media interface, and heterogeneity of porous media as well as the deformation of fibers and their interactions with the fluid flow are left for future studies.The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.


The Rose window instability is a little-explored electrohydrodynamic instability that manifests when a layer of low-conducting oil is placed in an electric field generated by corona discharge in a point-to-plane configuration. Above a critical voltage, the instability starts as a single dimple in the oil layer right below the point electrode and subsequently evolves into a characteristic pattern of polygonal cells. In this study, we experimentally explore governing parameters that guide the instability and document geometric attributes of the characteristic cellular pattern. The driving force for the instability has been attributed to the buildup of charged ions which in turn apply an electric pressure on the oil surface. We confirm the charged surface distribution using thermal imaging and demonstrate that the instability can be locally inhibited by preventing charge buildup under an ion shadow. Recently, it has been proposed that corona discharge can be an effective technique for controlled droplet spreading ,2. Coron2.a). The injection electrode is a sandpapered steel needle insulated on the body to keep away stray fields and corona leakage (figure 2b). The needle is hoisted on a burette stand and a vernier scale used to measure adjusted pin heights. The plate electrode consists of an FTO (fluorinated tin oxide) coated glass electrode [−3). The high voltage supply used for the experiments provides variable voltage in the range of 0–10 kV DC at 75–500 kHz. Voltage and current were measured using UNI-T UT60 and an EMCO V1G HV divider (1000 : 1). Schlieren photography using shadowgraphy was also done using a 5.1 inch parabolic mirror.Early research into the RWI  identififigure 2lectrode  which ai3.3.1.We investigate the critical voltage at which the RWI manifests itself as a function of the pin height and liquid layer thickness. Here, we define the start of the instability as the formation of a single dimple in the oil layer  which isL, the voltage V needed to obtain the same strength of electric field E (and correspondingly electric pressure P ∼ E2) must increase linearly in accord with E ∼ V/L as we can assume the electric field at the oil surface to be homogeneous due to a small ratio of the pin height and instability radius. Using an I–V graph like in To investigate the role of pin height, we start with an oil layer of thickness approximately 1 mm which is subsequently held constant. Next, we increment the pin height from 1 to 5 cm (using a step of 1 cm) and measure the voltage at which the instability initiates. We find that the critical voltage increases linearly with an increase in pin height . This trd (d within a narrow range of values from d ∼ 0.6 to 1.6 mm. Here, we explore a much larger thickness range from d ∼ 1.9 to 6.8 mm. The thickness was gradually increased by adding a certain amount of oil and measuring the depth using a vernier caliper. The pin height was kept constant at 2.8 cm. We find that even for the extended range of values the critical voltage V increases roughly linearly with d in keeping with incomplete results in fig. 7 of Pérez [Next, we investigate the role of the liquid film thickness d . Previouof Pérez , where tof Pérez .Figure 3.2.We explore geometric characteristics of the RWI for our point–plane configuration which, to our knowledge, has not yet been explored in detail. We observe that if the voltage is held constant, beyond the critical voltage, the unstable air–liquid interface settles into a stable configuration of polygonal cells with a near circular boundary. We explore the spatial extent of the cell pattern, measured as the radial distance from the pin to the edge of the cell pattern, which we refer to as the instability radius figures  and 8. WRadial heterogeneity in the electric field strength also manifests in the size of the polygonal cells that form due to RWI. In the tip–plane configuration, the electric field is strongest right underneath the tip and reduces radially away from the tip. Correspondingly, we find that the cell sizes vary radially with the smallest cells below the tip and larger cells at the boundary . Cell si3.3.The driving force for the RWI has been attributed to the action of the electric field on the surface charges that accumulate atop the low-conducting oil layer. As the charged ions bombard the surface, they carry a certain energy, part of which must be deposited onto the surface of the oil and over time should lead to a significant shift in the oil temperature. Thus, to reveal the distribution of these surface charges, we resort to thermal imaging. Using a thermal camera, we take images of the oil surface at four instances: one at the start, then at half a minute and a minute after the initiation of the instability and another after about 5 min . The rad3.4.Lastly, we investigate the evolution of the I–V curve in our experimental set-up. We note that the polygonal cells of the RWI form to allow for the transfer of the charged ions atop the oil to the grounded plate electrode. Thus, one might visualize the instability as a breakdown of the dielectric barrier of oil which prevents electrical conduction between the injecting electrode and the grounded plate. Our measured I–V curve confirms this notion. We find that before the instability initiates, we measure zero current at the grounded electrode. As the instability forms, ions are able to migrate to the grounded electrode and we correspondingly measure an increase in current. With increasing voltage, the depressions created by the RWI grow rapidly, allowing more ions to ground and correspondingly we record an exponential increase of the I–V curve.4.If we compare the RWI to other instabilities, we see that they have something in common, that is to achieve an equilibrium in the system. The RWI achieves a flow of current out of the system by essentially completing the circuit. It does that by creating the polygonal patterns. If we look at the case in b and c, where the central temperature increases by 0.5°C while border temperature remains rather unchanged.In figure 105.Instability of the air–liquid interface under an applied electric field has shown that the ions in effect of the electric field go towards the plane electrode. Unlike an air capacitor, in our point–plane configuration ions are produced which are charged. These ions flow towards the plate to be grounded but the dielectric layer in effect prevents them from doing so. Before the instability can occur, the charges accumulate on the surface of the oil and in effect of their relative energies, produce a certain force on the oil which when strong enough pushes the oil out of the way, This initial opening or weak spot in the oil is our initial dimple and this opening weakens further, increasing in size. The process of self-organization occurs in the oil to make the openings in essence the boundaries of polygons, which ensures stability of the system. As we see in


The Klyachko, Can, Binicioglu, and Shumovsky (KCBS) inequality is an important contextuality inequality in three-level system, which has been demonstrated experimentally by using quantum states. Using the path and polarization degrees of freedom of classical optics fields, we have constructed the classical trit (cetrit), tested the KCBS inequality and its geometrical form  in this work. The projection measurement has been implemented, the clear violations of the KCBS inequality and its geometrical form have been observed. This means that the contextuality inequality, which is commonly used in test of the conflict between quantum theory and noncontextual realism, may be used as a quantitative tool in classical optical coherence to describe correlation characteristics of the classical fields. Contextuality is a curious property in quantum world. In classical world, an observable is predefined and has a joint probability distribution when it is measured with the other observables. This observable is independent of other compatible observables that are measured simultaneously with it. This independency is called noncontextuality. Kochen and Specker proposed the original contextual theory in 196723456891011121417et al. using the qutrit encoded in three paths of the single photonset al.et al.et al. firstly proved the violation of the KCBS inequality with the single photon qutritet al.et al. have tested the violation of the geometrical form of the KCBS inequalityet al.30The state independent inequality , which have the values +1 or −1, the algebraic inequality for any choice of them can be expressed asAccording to ref. i + 1 modulo 5) are orthogonal, and they are compatible. When For the contextuality inequality, the observables φ〉 projects to the five couples of observables where φ〉 = |2〉. The geometric form of the KCBS inequality or Wright’s inequality is expressed as20where φ〉 is approximately 4π/15) can be obtained. Thus, we haveUnder the Kochen-Specker rules, the value at every direction vector is predetermined to 1 or 0, but the maximum is only one for the two orthogonal directions. Thus, the maximum is two for the five directions of pentagram. In quantum mechanics we select the same input state like that in the KCBS inequality, and The above inequalities  have bee25262728E0, E1 and E2 represent amplitudes of the classical optical fields, and Now we use classical optical fields to express the input state and the direction unit vectors for the KCBS inequality and its geometric form as showed in AiAi+1 of classical optical fields. In order to describe the KCBS inequality, ai) is the direction unit vector. For the input state |χ), from the correlation AiAi+1 as E2 need to be normalized. When it is normalized, the value In the following we calculate the correlations The geometric form of the KCBS inequality is also violated in the classical optical systems. In the following, we test experimentally the KCBS inequality and its geometrical form in the classical light systems.The test of the KCBS inequality needs a three-level system. Using classical optical beams, we can construct the corresponding system. The scheme is shown in E0, E1 and E2 represent the amplitudes of classical optical fields for the different paths. With tuning the HWP1 and HWP2, the different amplitudes of classical optical fields in three paths can be obtained, thus the desired input state can be represented with these bases, namely, where In the KCBS experiment, these observables correspond to their respective operators. After the input state is projected onto the eigenstates of each operator, the measurement for the observable can be implemented. In the three-level system, the operator owns three different eigenstates and the corresponding eigenvalues. Here the input state is projected onto the eigenstate and then the probability of the eigenvalue is obtained. When the eigenvalue is multiplied by its probability, we can obtain the observable by summing the product. Nevertheless, the probability can be obtained by detecting the light intensity in our classical optical experiments (see Methods section for details).Ai and Ai+1 need to be tested simultaneously. Here the operator ai) and P. The light intensity of each output port is normalized , and the probability of eigenvalue is obtained. The measurement probabilities for the output ports 1, 2 and 3 correspond to P, P and P, respectively, thus the observable A1A2 can be obtained.With suitably setting up the angles of HWP5, HWP7 and HWP8, the eigenstates of these observables can be obtained at three output ports. They meet the requirements of joint measurement. Taking Ai and Ai+1 are acquired, PPD1 = I1/(I1 + I2 + I3), PPD2 = I2/(I1 + I2 + I3) and PPD3 = I3/(I1 + I2 + I3). And then using the relation: The measurement methods for all five pairs of observables are the same. We tune the angles of the HWPs and obtain the desired eigenstates of the pairs of observables. The setting angles of the HWPs for the measurements of the different correlation pairs are given in Methods section. Then the input state is projected onto these eigenstates and the light intensities of the output ports are measured for every setting. The light intensity of each output port is normalized, therefore the joint probabilities of With summing the five sets of results, we can calculate the contextuality for ai)(ai| and implement the projective measurements in the measurement stage. As showed in ai), PPD2 = I2/(I1 + I2 + I3) and PPD3 = I3/(I1 + I2 + I3). With the probabilities we can calculate the value ai, which corresponds to the expression in Similarly we need to construct these eigenstates of the operator |The upper bound of the inequality for We have performed the experimental test for the KCBS inequality and its geometry form  in the classical light systems. Using the polarization and path, the cetrit has been constructed, the projection measurement has been implemented, the clear violations of the KCBS inequality and its geometrical form have been observed. This indicates that the contextuality inequality, which is commonly used in test of the conflict between quantum theory and noncontextual realism, can be used in the classical optical coherence to describe correlation characteristics of the classical fields. In addition, recent investigations have shown that there is a remarkable equivalence between the onset of contextuality and the possibility of universal quantum computationIn the classical optical systems, the direction unit vector can be expressed as Ai, and then we can calculate the observables. Here the eigenstates are also expressed with the classical optical fields, where |ai) is exactly the eigenstate corresponding to the eigenvalue −1 for the operator Ai (χ) is projected onto the eigenstate |ai). The probability |(ai|χ)|2 is the modular square of the amplitude of the optical field in the experiment, and the modular square of the amplitude is the light intensity IIn the measurement stage, the input state needs to be projected onto the different eigenstates of the operator Thus the probability of the eigenvalue can be expressed with the light intensity and we can detect the light intensity to obtain the probability of the eigenvalue in our experiment. The probabilities of two other eigenvalues can be measured similarly.θ is the rotation angle of the HWP. We assume that all amplitudes for the three input bases are unit amplitudes, and at the three output ports the output expressions may be written asFor the KCBS inequality, we need to construct the interrelated eigenstates that satisfy the joint measurement scheme at the three output ports. The transformation matrix of the HWP is ai).Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


Deep sequencing can detect somatic DNA mutations in tissues permitting inference of clonal relationships. This has been applied to human epidermis, where sun exposure leads to the accumulation of mutations and an increased risk of skin cancer. However, previous studies have yielded conflicting conclusions about the relative importance of positive selection and neutral drift in clonal evolution. Here, we sequenced larger areas of skin than previously, focusing on cancer-prone skin spanning five decades of life. The mutant clones identified were too large to be accounted for solely by neutral drift. Rather, using mathematical modelling and computational lattice-based simulations, we show that observed clone size distributions can be explained by a combination of neutral drift and stochastic nucleation of mutations at the boundary of expanding mutant clones that have a competitive advantage. These findings demonstrate that spatial context and cell competition cooperate to determine the fate of a mutant stem cell. Deep sequencing technologies allow for the investigation of clonal evolution in human cancers. Here the authors, combining sequencing data from human skin with mathematical modelling and simulations, suggest that the spatial context of a mutation with respect to other mutant clones may lead to differential clonal evolution. Typically, a specific or ubiquitous gene promoter is used to express Cre recombinase in the cells of interest and their progeny are fluorescently labelled for analysis. In human tissues, however, cell relationships must be inferred by other approaches. Historically, these have included the use of spontaneous mutations in mitochondrial and genomic DNA as clonal markers, in combination with analysis of methylation patterns in non-expressed genes3. More recently, deep sequencing has allowed the detection of hundreds of mutated genes and is being widely used to infer clonal relationships in a variety of tumour types5. One human tissue that lends itself to clonal analysis is the outer covering of the skin, the epidermis. The epidermis is maintained by cells that self-renew in the basal layer and differentiate in the suprabasal layers, forming a stratified squamous epithelium6. Skin is readily accessible in the form of surgical waste, and the techniques for whole-mount epidermal immunolabelling are well established7. Furthermore, the risk of skin cancer increases exponentially with age and is associated with accumulation of somatic mutations8. Genes that are frequently mutated in cutaneous squamous cell9 and basal cell10 carcinoma have been identified and can be used to infer clonal relationships. However, previous studies reveal a paradox, whereby there is evidence of positive selection of mutant epidermal clones11, yet clone size distributions are consistent with neutral drift14, a process by which the emergence of mutant clones is through genetic drift of mutant alleles that have neither a positive nor a negative effect on clone size.In mice, the use of genetic lineage tracing is a well-established technique for identifying subpopulations of cells that contribute to tissue homeostasis and disease15. Subsequently it was demonstrated that mutant cells can have a competitive advantage over neighbouring cells16 and that cell competition can play a physiological role in the regulation of cell populations19. We hypothesised that a similar mechanism may contribute to the differential survival and proliferation of mutant clones in the epidermis.One potential solution to this paradox is that there is competition between mutant cells. Cell competition is an evolutionarily conserved mechanism that leads to the outgrowth or elimination of relatively less fit cells from a tissue by competition with fitter cells. It was initially described in the developing Drosophila epithelium, where mutant cells are at a competitive disadvantageHere we reasoned that our understanding of clonal relationships and the potential role of cell competition in sun-exposed human skin could be improved by analysing more and larger samples than previously, by extending the analysis to skin from older individuals, and by sampling skin from donors who were at elevated risk of developing skin cancer. These approaches have led us to discover that clone size cannot be explained solely on the basis of neutral drift, but is also influenced by the spatial location of cells that acquire secondary mutations.20  samples from 10 patients aged 33–87 undergoing Mohs micrographic surgery for non-melanoma skin cancer9 and basal cell10 carcinoma . This implies that certain clones must exhibit a selective advantage and is consistent with mouse models showing that experimental inactivation of the Notch pathway in the basal layer of the mouse oesophagus26 or hedgehog signalling in mouse epidermal stem cells27 leads to the expansion of contiguous clones that have a competitive advantage over neighbouring stem cells.The stochastic division and differentiation of stem cells within the basal layer of the epidermis gives rise to a process of neutral drift, whereby a minority of clones expand and others are extinguishednts Fig.  and whennts Fig. . HoweverTo understand the large size of the clone size distribution, we developed a lattice model of stem cell behaviour in a homogenous two-dimensional stem cell compartment Fig. . We next30 indicate that the rate of stem cell migration is negligible in the basal layer of the epidermis in the steady state. This is supported by lattice-based modelling, which demonstrates that a non-negligible rate of stem cell migration leads to dispersed, scattered clones , this does not alter the clone size distribution  reveals that a model analogous to that described above will also arise for clones growing exponentially, in this case, giving rise to an inverse power law distribution with squared rather than cubic dependency. A comparison of the fit of our data to inverse squared and cubic power laws is shown  gives an inverse cubic power law dependency for the distribution at larger clone sizes. In some circumstances, small clones appear to undergo a transient period of exponential growthown Fig. .31. In human epidermis stem cell clusters expressing high levels of β1-integrins are surrounded by cells with more limited proliferative potential, which have a high likelihood of initiating terminal differentiation and expressing keratin 10  cells with limited proliferative potential . This leads to the expansion of mutant stem cell clones into areas previously occupied by TA and wild-type stem cells  promotion of cell death of adjacent wild-type cells39. There is evidence for such a mechanism in the context of disrupted Notch signalling both in vitro29 and in vivo27. Furthermore, expansion of preneoplastic APC mutant clones in a Drosophila model40 appears to follow this paradigm. Our experimental data are not able to differentiate the effects of cell autonomous and non-cell autonomous mechanisms, since the consequences for our models are the same, namely a local competitive advantage for cells at the boundary of the expanding clone.While the conclusions of our study are of particular relevance to disruption of Notch signalling, they are potentially applicable to any mechanism that leads to a competitive advantage for the mutant clone. This concept of cell competition was initially recognised in the developing Drosophila wing epitheliumOur conclusion, that the fate of a mutant stem cell is influenced by a combination of neutral drift, cell competition and by the spatial context in which it arises, has important implications for the interpretation of sequencing studies of preneoplastic tissues and for understanding the earliest stages in the development of malignancy. It will be interesting to discover whether spatial context is also important in the evolution of mutant clones in other epithelia, such as the intestine, and whether it is a general feature of cancer-associated mutations or is specific to the effects of UV light. We also speculate that, for any given combination of genetic lesions, spatial context will influence the likelihood that they will contribute to tumour progression.Anonymised skin samples samples were obtained from 10 patients of 33–87 years of age with Fitzpatrick skin type I-II undergoing Mohs micrographic surgery for the treatment of non-melanoma skin cancer affecting the head and neck. The project was subject to ethical approval by the research tissue bank at St John’s Institute of Dermatology (Guy’s and St Thomas’ NHS Foundation Trust) and patients gave informed consent for the inclusion of their tissue samples. Samples were collected from excess skin removed at the time of reconstruction of the Mohs defect following histological clearance of the tumour, thus we can be confident that there was no extension of the tumour into the collected skin specimens. Skin samples were flash-frozen and subsequently stored at −80 °C prior to further processing. In addition, salivary samples of 2–3 ml saliva were obtained from these 10 patients for the extraction of genomic control DNA.Following retrieval from the research tissue bank, fresh frozen skin samples were thawed at room temperature. Samples were trimmed to an area of 4 × 4 mm. It is noted that, irrespective of donor site, skin samples will contract after excision due to innate skin elasticity and there may be differences in the degree of contraction according to donor site. Excess skin was employed for the preparation of paraffin blocks. Haematoxylin and Eosin stains were prepared from all samples. Skin samples from which DNA was to be extracted were incubated at 52 °C for 10 min in phosphate buffered saline (PBS). Following this treatment, the epidermis can readily be separated from the papillary dermis. To ensure that the entire epidermis was collected intact, separation was visualised with a dissecting microscope.To analyse the large scale clonal architecture of the samples DNA was extracted from the entire 4 × 4 section of epidermis using the Qiagen DNeasy blood and tissue kit according to the manufacturer’s instructions. Salivary samples were washed in PBS and subsequently DNA was extracted in an identical manner.CDKN2A was included in the original capture design, capture failed across all samples and therefore it was excluded from all further analysis. Capture was performed according to the SeqCap EZ Library SR User’s Guide  and library preparation was performed according to the TruSeq DNA Sample Preparation Guide  with no modifications.A custom capture strategy (NimbleGen SeqCap EZ) was designed with capture oligos targeting 121 genes frequently mutated in non-melanoma skin cancer  and subsequently mapped with Stampy42 run with default parameters then sorted with Picard. Sequencing was performed on the Ilumina Hiseq2000 or Hiseq 25000 with TruSeq SBS v3 chemistry. Average per-sample coverage was 1000×, giving overall rates of coverage across the 10 normal and matched salivary control samples of 20,000×. PCR duplicates and reads with a Phred quality score of <30 were removed from the analysis.Reads were pre-mapped with BWA11. Since we did not subdivide the epidermal DNA specimens from each individual, we instead constructed the background error model across all of the control .The identification of subclonal somatic mutations in high throughput sequencing data is challenging due to the error rate that reflects a combination of sequencing and PCR artefacts. This error rate varies according to genomic location, and furthermore, whereas a genuine mutation is anticipated to be present equally on both strands, a PCR error will preferentially affect one strand. We employed statistical methodology identical to that reported by Martincorena et al.j and each normal salivary DNA samples i, Xijk, k ∈  with—corresponding to deletion of the base. The background error rate was fit to a beta binomial distribution  Notch1 was performed with a commercially available antibody . Tissue samples were embedded in optimal cutting temperature compound (Life Technologies) directly or fixed first with 4% paraformaldehyde for 20 min at room temperature, and stored at −80 °C. Ten to sixteen micron sections were cut using a Thermo Cryostar Nx70 (Thermo Fisher Scientific), Sections were transferred to blocking buffer for 1 h at room temperature, labelled with primary antibodies diluted in blocking buffer overnight at 4 °C, washed in PBS and then labelled with secondary antibodies and DAPI for 1 h at room temperature. Fluorescence mounting medium (DAKO) was used for mounting the coverslips (SLS) on the slides.43. Breast or abdomen skin samples were obtained as surgical waste with appropriate ethical approval. Small pieces (2 cm2) of whole skin were treated with Dispase (Corning) overnight on ice at 4 °C. The epidermis was separated as an intact sheet from the dermis and immediately fixed in 4% paraformaldehyde for 1 h. Epidermal sheets were then washed and stored in PBS containing 0.2% sodium azide at 4 °C until staining. Sheets were permeabilised and blocked using PB buffer  for 30 min at 37 °C in a shaking incubator at 100 r.p.m. Epidermal sheets were then stained with specific primary antibodies  in PB buffer overnight at 4 °C in a 24-well tissue culture plate. Samples were washed and incubated with corresponding secondary antibodies for 2 h in room temperature and washed before mounting on a slide with ProLong anti-fade mounting medium (Thermo Fisher) with DAPI  to reveal nuclei. A Nikon A1 confocal microscope was used for image acquisition. NIS Elements (Nikon Instruments Inc.) software was used for 3D maximal projection (1024 × 1024 dpi), volume rendering and deconvolution on z-stacked whole-mount images.Human epidermal whole-mounts were prepared and labelled using a procedure that was modified from previous reportsn at time t assuming neutral drift is given by12:rλ is the rate of stem cell loss/replacement in the basal layer. Thus, for a stem cell population of size N, the cumulative probability of a clone of size s or greater isWe have employed three approaches to understand the maximum clone sizes that are predicted to arise by neutral drift within the timespan of a human lifespan. Firstly, the size distribution of mutant clones of size 2 with an estimated basal cell progenitor cell density of 100 cell per mm, gives a total cell population of 4002. Assuming a rate of loss/replacement λ in the basal layer of 0.5 per cell per week, the probability of observing the largest clones in our data set, which comprise 64,000 cells  gives a cumulative probability of 4.4 × 10−13. Clearly, it is highly improbable that clones of this size will be observed in vivo.for a basal cell area of 4 mm44 derives the time to saturation T for a lattice comprising two cell types in neutral competition (r = 1/2).N = 64,000 cells (our largest clones) the time to saturation was in excess of 29,000 years; a 1 mm2 clone would require >2900 years according to this model.Secondly, with analogy to a classical voter model of surface catalysis, Kleinλ = 0.5 week-1 with the stochastic occurrence of neutral mutations. The largest observed clones comprised a fraction of 0.025 of the total population  for 100 years with In order to compare the clone size distribution observed in our data with the predictions of a model of neutral drift, we employed the formulation developed by Simons (Equation 2)n the mutant clone size, rλt a decay constant corresponding to the average size of a mutant clone induced at the time of first exposure to a mutation. In an identical manner to the plots presented by Simons, clone size distributions were fitted to two parameters: rλt, and n0—a correction for the size limit of high throughput sequencing for the detection of mutant clones.The left hand side represents the first incomplete moment and the the right hand side represents the exponential form assumed by this distribution, where a cellular population is subject to cell loss and replacement under the assumptions of neutral drift. Where We now describe a geometric formulation of the boundary nucleation model of clonal expansion. While this geometric formulation omits stochastic factors and other effects of clonal competition that are modelled by the lattice simulation, it does illustrate important concepts that govern the emergence of the observed clone size distribution.ω is the neutral mutation rate and θ the non-neutral mutation rate. A primary clone arises as a single cell and expands outwards as a circle. We assume that subsequent mutations are neutral and occur stochastically with rate ω for all cells in the stem cell compartment. Neutral mutations arising in wild-type cells are subject to neutral drift, similarly neutral mutations arising in mutant cells within the centre of the clone are also subject to neutral drift. However, if a neutral mutation occurs in a cell at the boundary of the mutant clone there is the possibility that this mutation will be amplified as a passenger along with the mutant clone as it expands displacing surrounding normal cells. As the size of the primary clone increases, the probability of nucleation of a secondary daughter clone containing an additional neutral mutation increases in proportion to the number of cells at the boundary; however, the potential size to which this secondary clone can expand decreases due to both competition with other cells at the boundary and since the primary clone is closer to its maximal size. Since clones arising via this mechanism are far larger than those arising by neutral drift within the same timescale, the latter are omitted from the model. This model is compatible with both expansion terminated by collision between competing mutant clones or with an extrinsic process that limits expansion.We consider the case of a primary clone carrying a non-neutral mutation, which leads it to outcompete surrounding wild-type stem cells in a two-dimensional stem cell compartment such as the basal layer of the epidermis. We assume that T, we observe a primary clone of radius σ is the diameter of a single cell. This primary clone has expanded from a single cell in the period between t0 and T. At time t, the radius of the primary clone is r and the number of secondary clones N that initiate at the boundary at t is proportional to the number of cells that comprise the circumference,γ is a constant determined by the neutral mutations rate ω and the nature of the competitive advantage conferred unto the primary clone. Although some secondary clones will be extinguished and others will grow to larger size, on average, we assume that the area achieved by a secondary clone is given byAt time r ≈ σ, where stochastic factors will dominate and has no physical meaning where 2r < σ. Rearranging to get r and taking the positive-signed solution of the quadraticIt is noted that this model will not accurately represent the situation when M, the number of clones of size A as a function of AAmax is the size of the largest secondary clones. Taking the derivative of r with respect to AIntegrating over time and changing variables gives r(t) = αt as a consequence of the competitive advantage of mutant over wild-type stem cells acting at the boundary. Substituting:We assume that the primary clone expands as β, by taking the gradient of the logarithm of the distribution against the logarithm of area,Our experimentally observed clone size distributions appear linear when plotted on a log–log scale Fig. . This isA gives β(inf) = −3. Thus, the distribution approaches an inverse cubic power law quickly as the size of secondary clones becomes non-negligible with regard to the primary clone. We can also derive this power law dependence from the leading term as σ tends to a small number, keeping R constant:taking the limit for large 26.The lattice simulation and geometric model described above assume that all growth of the primary clone arises from division of cells at the boundary. As a consequence, the maximum rate at which the clone can expand is proportional to the number of cells at the boundary, i.e., the circumference of the clone. This gives a squared dependency for total clone size (area) on time. However, in some circumstances, it appears smaller clones can undergo a transient period of exponential growth26.For exponential growth, the rate of clonal expansion must be proportional to the total number of cells in the clone, implying that all cells in the clone have an equal probability of dividing. For a bacterial colony in suspension culture, this situation can arise since there is no spatial limitation on division, however, for a clone in a two-dimensional stem cell compartment, the maximum rate of growth is determined by the rate of flux of mutant cells into regions previously occupied by wild-type stem cells. This rate will be limited by the length of the boundary, the maximal rate of cellular migration and the rate at which neighbouring wild-type stem cells can be displaced. Since the total number of cells in the mutant clone increases as radius squared, whereas the length of the boundary increases proportionate to the radius, this flux must inevitably become rate limiting and this may explain why small clones can show exponential growth initially but rapidly revert to a slower growth rate at larger sizesθ is the rate of non-neutral mutations that confer a competitive advantage and ω the rate of neutral mutations. It is straightforward to derive a model for the expected distribution of large  mutations assuming exponential growth since both primary and secondary mutations must expand at the same rate and the rate of secondary clone initiation is proportional to the number of cells in the primary clone.Assuming that mutant clones can expand exponentially in vivo, at least transiently, let us consider the implications for mutant clone size distributions assuming an analogous mechanism to that described for the boundary nucleation model. As above, we assume that AP containing a non-neutral mutation at time T that has expanded exponentially to size t0 the area of the primary clone AP at time t is given byσ is the diameter of a single cell. From the time that they are nucleated, secondary clones must grow in proportion to the primary clone, and therefore at the final time T, the area of a secondary clone A is determined by the time of nucleation tIf we observe a primary clone RearrangingWe will needThe rate of new clone nucleation is proportional to the area of the primary cloneSowhich trivially has a power law dependence of inverse squared area.Li for which each site i is occupied by either a wild-type stem cell denoted by An, where  or is empty, denoted by L, is subject to the following transitions  descended directly at from An at time t and all earlier time points. Thus, the clonal history of the model is represented by a tree with root node A0 and the size of clone n at time t is given by a recursive sum of clone n and all children of clone n at time t:〈This methodology can represent clone bifurcation, whereby a subsets of the clone acquire different subsequent mutations and can thus accurately model the sequential accumulation of unique mutations within the genome.L is implemented by stochastically swapping the contents of adjacent cells with rate rκ:Lj,t ∈ Ni,tCellular migration on the lattice L with a rate different from rλ. Other factors include the possibility that a member of the set Ni,t has a probability of filling a vacancy Bn alters the behaviour of other cells on the lattice (non-cell autonomous effects). Outside the context of this lattice model it is possible that mutant stem cells could alter the size of the stem cell compartment or altered mobility within the basal layer. For the purpose of this simulation, we are primarily interested in the fixation of secondary neutral mutations during expansion of the primary mutant clone and the precise nature of the replicative advantage is less important therefore we focus on the former scenario.Next, we consider the situation where mutations arise which alter neutral dynamics. In the context of the lattice model the most important of these is the probability that a stem cell is lost from We allow non-neutral mutations to arise stochastically as follows:B and acquires a new clonal label m. In most circumstances, it is anticipated that rλ in cells of type A and rϕ for cells of type B. Thus, the lattice transitions governing the model of non-neutral dynamics are as follows:A stem cell with non-neutral dynamics is denoted by C to represent a TA cell. C is subject to the following lattice transitions:-rν is the rate of differentiation of stem cells into TA cells, n is the clonal label associated with the cell, and q is the total number of replications completed by the TA clone. ν is a function of location on the lattice Li: ν = diff(i). While diff(i) can take any form, for simplicity we constrain it to a binary output with biologically representative forms, whereby stem cells are grouped in clusters as in the basal layer of the human epidermis. To limit the proliferative potential of TA cells a maximum of qmax replications are permitted.To further extend the model to a represent a heterogenous stem cell compartment containing both stem and TA cells, we introduce a novel cell type E:A is the set of all stem cells, C the set of all TA cells, p an arbitrarily large scalar penalty. This function takes higher values, where stem cells are admixed with TA cells or where TA cells are localised within an area that contains a stem cell cluster. For two neighbouring sites on the lattice, cellular migration is only permitted where this leads to an overall reduction in lattice energy.Experimentally, we observe that stem cells and TA cells cluster together, furthermore TA cells are rarely seen within the TA cluster. To model this tendency, we define a lattice energy function The lattice models described above are not readily amenable to analytic approaches and therefore we proceeded to computational simulation.rλ has been estimated as 0.5 per week12 and the simulation was implemented with discrete quantised time intervals of 1 day. All lattice transitions were assumed to initiate and complete within a single time step. A large majority of new clones created are lost from the lattice as a result of neutral drift, and therefore, in order to prevent the memory requirements of the clonal history of the simulation from growing to an excessive size, it is necessary to periodically purge the clonal history of clones for which no members of that clone or its descendants survive.The lattice model was implemented as a computational simulation on a hexagonal lattice with dimensions 200 × 200 or 400 × 400 (Supplementary Methods). The rate of stem cell loss/replacement https://github.com/latticesim/sim. With the exception of simulations taking an excessive amount of computational time, all simulations were run a minimum of 10 times with different random seeding and results are presented as mean ± standard deviation.The model was implemented in C++ and Python. Source code is available at the following URL: A full list of identified mutations is provided (Supplementary Data Supplementary InformationPeer Review FileDescription of Additional Supplementary FilesSupplementary Movie 1Supplementary Movie 2Supplementary Data 1


Anthropogenically induced climate change has created a set of intriguing scientific problems pertaining to the Seas and the Oceans on earth that can be monitored, analysed, modelled and consequently understood at some level. The international research community is deeply engaged in this endeavour as exemplified by the United Nations Decade of Ocean Science for Sustainable Development (2021–2030). Even so, the types of problems caused by human activity are inherently difficult to solve, if solvable at all, making them equally difficult to approach and manage. Here we address these as ‘wicked problems’  will alter the entire marine food web through trophic cascades, including pelagic primary production, and might have unforeseen impacts on the general energy flow of the entire marine ecosystem , and scenarios suggest that the trend is steepening .On a geopolitical level the UN’s 17 Sustainable Development Goals are important, embodying a moderate progress. They are written in the expectant language that international politics demands. Goal #14, curiously dubbed ‘Life 2 emissions, ocean acidification is easier to grasp, but harder to reverse. More CO2 in the atmosphere due to combustion of fossil fuels equates with more uptake of CO2 by the Oceans , and once the CO2 is absorbed by the upper layer of the ocean it cannot easily be undone.As opposed to many other consequences of industrial COThere is no backdoor out of this wicked problem, which we barely understand the range or severity of. There is no silver bullet that will present a universal remedy for the wickedness. Having a political target that depends upon ‘enhanced scientific cooperation at all levels are’ is courteous to the science community, but will do little to ‘minimize the impact’ or, more importantly, halt the current development on human time scales unless other agents and processes put that insight to good use.The Montreal Protocol on Substances that Deplete the Ozone Layer, only for the ocean, would serve the purpose only if it were fast enough. Precisely how The Paris Agreement will be played out in terms of reduced CO2 emissions remains to be seen, as is true for the COP23 meeting in Bonn in November 2017, but atmospheric concentrations of CO2 will continue to soar in decades to come and have in fact increased for 2017 as well (Le Quéré et al. Acknowledging the problem, making it emblematic, but doing next to nothing is confusing and demotivating. It paints a contradictory picture of reality that obfuscates a clear understanding of what the crux of the problem is: our continued emissions of greenhouse gases, our continued harvests and usage of ecosystem goods and services. And by doing so, it becomes increasingly harder for the public to identify pathways to action that at least resolve parts of the problem. We need an opening that shows the way forward, and we need positive and sustainable examples based on scientific knowledge, implemented at full scale rather than just demonstration-projects. An equivalent to Global sea level rise is another wicked problem. There are many others including deoxygenation (Keeling et al. Consider the fact that since the mid-20th century over 90% of the excess heat in the atmosphere due to greenhouse gases has been absorbed by the Oceans and during the last century or so surface waters have warmed by 0.7° ± 0.08 °C (Jewett et al. The Oceans and the Seas are sometimes posed as a potential redeemer of the perilous climate and environmental challenges we are currently facing only if we treat it right Cf. OECD —a messagWe need serious and solid science-and-society interactions to grasp, comprehend, communicate and ultimately perhaps try to tame some of the wickedness of the anthropogenized Ocean. A step in the right direction would, according to Frame , be to b


Hyperelastic finite element models, with either an idealized cylindrical geometry or with realistic craniectomy geometries, were used to explore clinical issues relating to decompressive craniectomy. The potential damage in the brain tissue was estimated by calculating the volume of material exceeding a critical shear strain. Results from the idealized model showed how the potentially damaged volume of brain tissue increased with an increasing volume of brain tissue herniating from the skull cavity and with a reduction in craniectomy area. For a given herniated volume, there was a critical craniectomy diameter where the volume exceeding a critical shear strain fell to zero. The effects of details at the craniectomy edge, specifically a fillet radius and a chamfer on the bone margin, were found to be relatively slight, assuming that the dura is retained to provide effective protection. The location in the brain associated with volume expansion and details of the material modeling were found to have a relatively modest effect on the predicted damage volume. The volume of highly sheared material in the realistic models of the craniectomy varied roughly in line with differences in the craniectomy area. Traumatic brain injury (TBI) can cause swelling in the brain leading to uncontrolled raised intracranial pressure (ICP), which in turn can cause death or severe brain damage. Hence an important factor in treatment of TBI is reduction of ICP, which can be achieved either by medical or surgical therapies.A number of clinical issues arise which it is hoped that a bioengineering approach can help address.et al.There are two standard forms of DC, bifrontal, and unilateral craniectomies, details of which have been reviewed by Timofeev Related to the location of the craniectomy, the surgeon has to consider the optimum size of the opening. Evidently a larger opening is likely to lead to a better reduction in ICP, but at a clinical cost of increased risk of adverse effects and complications associated with the surgery.et al.Historically, surgical technique suggests creating a positive chamfer on the bone edge in order to stop the bone flap sinking when it is replaced. Modern cranioplasty tends to be carried out with a titanium or synthetic plate which overlaps the edge of the craniectomy, raising the possibility of using no chamfer, or even a negative chamfer, as illustrated in Fig. et al.et al.,There has been a significant body of work to address brain biomechanics, as reviewed by Goriely More recently Li and HolstIn parallel with advances in biomechanics models of tissue deformation, there has been significant work improving the extraction of patient-specific geometries for more accurate modeling of deformation behaviour.via FE modeling. A specific focus of the work is estimating the volume of the brain which might be at risk of damage post operation, by determining the region of brain tissue exceeding a critical strain associated with cell death.The aim of this paper is to follow up on the above modeling work, addressing the clinical questions described above Δ of a rigid platen at the top of the cylinder. This platen contains the circular craniectomy of radius a. The platen makes frictionless contact with the brain tissue. As the cylinder reduces in volume, brain tissue is forced out of the craniectomy hole. Frictionless contact conditions between the skull and brain tissue on the top face and sides of the cylinder were simulated by constraining brain motion on these faces to be only in the radial and axial directions, respectively, whilst the bottom face was fixed to the skull.The skull and craniectomy were modeled in the finite element software Abaqus  by an idealized geometry of a cylindrical body containing a circular craniectomy, as illustrated in Fig. πR2. The maximum bulge deformation δ of the tissue at the herniation is found from the difference in heights between the bottom of the platen and the surface of the brain at the centre of the craniectomy, Fig. The independent variable in the problem was considered to be the herniated volume of material, which is given by the platen displacement Δ times the cylindrical cross-sectional area The loading arrangement is intended to represent the clinical situation, where swelling of the brain and an excessive ICP provides the driving force for the herniation of the brain out of the craniectomy opening. Because of the frictionless contact conditions, it is not important where the reduction in cranial volume is generated. Moreover, it is supposed that the details of the strains at the craniectomy will be equivalent for the clinical situation of the enlargement of brain volume, and the modeled situation of a reduction in skull volume. This importance of this assumption, and of the location of the region at which swelling occurs, is addressed in the paper.ρ was included on the inside of the skull at the edge of the craniectomy to simulate the real situation where the dura acts to smooth the corners of the contact between the brain and the skull. The dura is a stiff membrane around the brain. During the DC procedure the dura is circumferentially opened, but the dural opening is smaller than the bone craniectomy opening. This allows the dura to protect the brain from any sharpened edges of the surrounding bone as the brain expands outside the skull. Given that the dura ranges in thickness from 0.3 to 0.8 mm depending on age,et al.,A fillet radius The model was meshed using axisymmetric 8-noded hybrid elements with reduced integration. The mesh was refined near the craniectomy edge, with a minimum mesh size of 0.05 mm. The resulting model contained approximately 50,000 elements. A mesh refinement study, detailed in Fletcher,An implicit analysis was used to solve the problem, including large deformation analysis. Results were analysed using Matlab .An advantage of the idealized model is that it is simple to vary key geometric parameters systematically. Changes in craniectomy diameter, fillet radius, and chamfer angle were considered, varying these parameters in turn from a baseline case with a craniectomy diameter of 10 cm, an internal fillet radius of 1 mm, and with no chamfer  gap between the outer surface of the brain and the inner table. This ensures that there were no overlapping elements between the skull and brain prior to the subsequent expansion of the brain. Moreover a soft exponential contact condition was used to aid convergence, choosing parameters for the clearance and typical pressure of 0.5 mm and 1 kPa, respectively.Uniform expansion of the brain was generated by changing the temperature of the entire model during the load step, relying on thermal expansion to give the required change in volume. The actual temperature of the model is not part of the solution procedure; the temperature change is solely used as a device to generate the volume change. Due to convergence issues, the herniated volume achieved in these cases was only 22 mL, significantly less than a typical volume clinically.The extraction of the volume of material region where the shear strain exceeded a critical was performed in Abaqus viewer, using the display group function to isolate and sum the volume of elements which had shear strains greater than the threshold.Four craniectomy geometries were created in the FE model, the two more common unilateral and bifrontal geometries, and the less common bilateral and bifrontal with a midline bar of bone.Points defining the location of the craniectomy were placed on the surface of a three-dimensional render of the MRI dataset used to create the FE model by one of the authors (AK), who is a practising neurosurgeon. These points were used to define a cut through the skull in 3D Slicer,et al.The locations of the craniectomy openings are illustrated in Fig. A potential limitation of the idealized model is the absence of internal features including the ventricle, and the simplified way in which the expansion has been modeled by a reduction in skull volume. To investigate these effects, further FE models were developed, taking a 10 cm diameter craniectomy and uniformly expanding the brain tissue, rather than reducing the “skull” volume. To explore the effect of ventricles on the deformation, a “ventricle” of approximately elliptical cross section, with initial volume of 29 mL and diameter of 6.4 cm was created at the centre of the brain. This ventricle was ascribed a shear modulus 1/30th of the surrounding material. To understand how the deformation depends on the location where expansion occurs, a model was considered where the central ventricle was given an expansion corresponding to 50% of the total 85 mL herniation, with the remaining 50% contributed by the remaining brain tissue.et al.via a bulk modulus term D1. Values of D1 equal to 7.2 and 93 kPa were used, corresponding to effective Poisson’s ratios of 0.496−10 m/s was used.Another potential limitation of the work is the simplified hyper-elastic material used. To quantify the sensitivity of the results to the inclusion of time-dependent deformation behavior associated with poro-elastic effects, a poro-elastic FE model has been developed for the idealized case, following the work of Fletcher To make a comparison with the idealized geometry modeling, in-plane deformations were derived from clinical CT X-ray images pre- and post-op. To obtain the clinical deformation field, the pre-op CT scan of a patient with TBI was mapped to the post-op CT scan after DC using the BRAINSDemonWarp moduleAll CT scans were acquired during routine clinical care. Anonymised clinical data were collected in the course of the RESCUEicp study , RESCUE-ASDH study , and from clinical audit of patient care in the Neurosciences Critical Care Unit/Neurosurgical Unit of Addenbrooke’s Hospital. Ethical approval for the RESCUEicp study has been obtained from the UK Multi Centre Research Ethics Committee (Eastern Region) and the clinical audit has been registered and approved by the Clinical Audit Department, Addenbrooke’s Hospital. The RESCUE-ASDH study acquired ethical approval from the National Research Ethics Service (North West Region). All records/information were anonymized and de-identified prior to analysis.http://dx.doi.org/10.17863/CAM.14.Data relating to the study are available at Cambridge University’s online repository at Figure 2, where the area was characterized by the height times the width of the craniectomy.Figure The shear strain within the brain tissue is plotted in Fig. Figure The effect of craniectomy diameter on the shape of the herniated brain surface and the contours of critical strain are shown in Fig. The evolution of the damage volume with herniated volume is shown in Fig. Results for the effect of craniectomy diameter on damage volume are summarized in Fig. Figures Figures Figure et al.Figure Figure Table Although the cylindrical model does not have the same geometry as a skull, the trends seen between geometric parameters and the region at risk of damage are expected to apply to the clinical situation, albeit with quantitative differences associated with a realistic geometry.Results from the idealized model confirm the expectation that a larger craniectomy opening is potentially less likely to damage tissue due to high strains present after the DC procedure. However, for a given herniation volume, there is a threshold area beyond which the volume of damage falls to zero, so that a further increase in craniectomy area would not be predicted to give a significant improvement in damage area. For example, for clinically typical herniation volumes of 50 and 100 mL, this threshold diameter is around 10 and 13 cm, respectively, Fig. The qualitative comparison of deformation between the idealized model and a clinical case, Fig. et al.,via increasing the ventricle volume. But clinical observations show that such ventricle volume changes do not fully account for the herniated volume observed. Hence there is a need for a significant additional mechanism of volume increase, which in von Holst and Liet al.Another potential limitation of the work is the simplified hyper-elastic material used. The approach used here has been supported by the preliminary FE study of idealized DC geometry by Fletcher In any case the results of Fig. An investigation of the effect of craniectomy edge fillet radius and chamfer suggest that, as long as the dura provides an effective protection from the sharp edge of the skull, the details of the craniectomy edge are not critical to the area of potential tissue damage. There are relatively small changes in shear strain local to the craniectomy edge, but these are over-shadowed by the much larger volumes of potentially damaged material sub-surface. There was no significant change in deformation associated with the skull fillet radius changing from 0.5 to 2 mm. In terms of chamfer at the craniectomy edge, the best situation was seen to be the 90° chamfer, corresponding to a cut perpendicular to the skull surface. The sharp 45° chamfer gave a small region of heightened shear strain compared to the 90° chamfer, albeit the change was relatively small. The worst case was the 135° chamfer which, although providing a more gradual corner at the inner surface, resulted in a higher area of shear strain and effectively reduced the craniectomy area.In general the realistic geometry results correlate with the results of the idealistic model, with the larger craniectomy areas giving smaller deformation and smaller potential damage regions. An exception to this is the bifrontal with the midline bar geometry, which had a somewhat higher volume of high shear strain than might be expected given its area. This is associated with the constraint of the bony bridge over the sagittal sinus which blocks the expansion of the brain in the region of largest deformation.in vitro model of TBI utilizing stretch of organotypic slice cultures of the rat cortex.The strain threshold used in the present study was based on an Due to convergence issues, the expansion of the brain for the realistic model was restricted to 22 mL, on the low side compared with typical clinical values in the range 20–140 mL.the maximum bulge displacement is predicted to increase roughly linearly with increasing herniated volume, also decreasing with increasing craniectomy area for a given herniated volume;the region of maximum shear strain is below the surface of the brain, increasing in extent with increasing herniation volume and decreasing craniectomy diameter;for a given herniation volume, there is a threshold craniectomy diameter at which the volume of potentially damaged tissue falls to zero;the effect of details at the craniectomy edge (fillet radius and chamfer) are relatively small, assuming that the dura is retained to provide effective protection;the volume of highly sheared material in the realistic models of the craniectomy varies roughly in line with changes in craniectomy area, although the case of the bifrontal with midline bar is notable for giving somewhat more volume of damaged material than would be expected from the craniectomy area;while the above models provide biomechanical guidance for changes in DC geometry, these will need to be balanced with clinical issues.Hyperelastic finite element models, with either an idealized cylindrical geometry or with realistic craniectomy geometries, have been used to explore clinical issues relating to DC. The potential damage in the brain tissue has been estimated by calculating the volume of brain tissue exceeding a critical shear strain. The following conclusions can be drawn:


Tipping points in complex systems are structural transitions from one state to another. In financial markets these critical points are connected to systemic risks, which have led to financial crisis in the past. Due to this, researchers are studying tipping points with different methods. This paper introduces a new method which bridges the gap between real-world portfolio management and statistical facts in financial markets in order to give more insight into the mechanics of financial markets. A striking effect in observed price time series are sudden trend switches5, which sometimes affect only a few assets, while at other times a whole market is impacted and moves in a synchronised fashion. Such turmoils are not limited to stock markets but might have ripple effects on the economy of a country or even of the whole world6.Understanding the links between different assets, economic sectors or even between economic zones gives the opportunity to uncover the internal structures of financial markets. These interactions are part of the complex mechanics, which drive prices at stock exchanges up or downtipping points are an interdisciplinary phenomenon found in fields with complex networks e.g. physics, biology and climate research8. Scheffer et al.8 describe two possibilities of a regime shift. The first results from an exogenous shock while the other one is endogenous. Especially in financial systems connectivity and homogeneity play a major role in crisis research. A small perturbation can have a cascading effect threatening the whole system. When approaching an endogenous tipping point one can observe critical slowing down, meaning a perturbation to the system state takes a long time to revert back to the preferred local state.These Hence it is of great interest to develop tools in order to identify current market states.19. It has been found that the role of the market mode in emerging markets is more important than in developed economic zones where in comparison the clustering of sectors is more dominant21.A common method at the disposal of researchers is the use of the correlation matrix between the log-return time-series of assets traded at stock markets. By using Random Matrix Theory (RMT) on such a correlation matrix it has been shown that markets consist of a dominate market mode and different economic sectors, which exhibit common trends22. In 2010, Billio et al.23 used Principal Component Analysis (PCA) on the correlation matrix derived from the return series with a sliding time window. They showed that for hedge funds, banks, brokers and insurance companies an increasing entanglement was present during the time of crisis in 200824. This showed the prominent role of these market participants.With the financial crisis of 2008 the focus of researchers shifted to the empirical measurement of systemic risket al.22 used the eigenvectors of the time dependent correlation matrix consisting of ten US sector indices. They found that by tracing the sum of the four largest eigenvalues over time, one can relate the steepest increase to the systemic risk of the US economy represented by the chosen sector indices. Furthermore, they outlined that the magnitude of an increase corresponds to the interconnectedness of the system and therefore provided a precursor indicator.Following this train of thought, Zheng In this paper, we introduce a new method which links known empirical facts to portfolio management research in order to empirically quantify market states in the S&P 500.26 and we assume that these investors are not noise traders27 but follow the principle of risk reduction by diversification30 and belong to the group of fundamental traders31. The general idea of a diversified portfolio is that it is more robust against turbulences since not all selected assets are influenced by the news regarding one particular sector32. However, determining the correlation between assets faces the problem that time-series in financial markets react on exogenous news and are therefore not stable36 as commonly assumed37. This forces investors to rebalance their portfolio to keep up with the changes within the market. One can imagine the inverse of rebalancing need as the energy barrier, which prevents an investor from moving to the next energy basin and a new preferred state. This paper aims to measure these critical transition points in time.We start with the assumption, that only a few big market participants are responsible for the major changes in share priceset al.38, who investigated the similarity between correlation matrices at different times, we define the similarity between two points in time t, t′ as the L1-norm similarity.ζ effectively measure the transaction costs an investor has to pay in order to rebalance the portfolio from one point in time to another. We use the daily closing prices of the S&P 500 components from 2000 until the end of 2016 as input data for the creation of mean-variance portfolios with a time window ω of 3 years.Extending the approach by Münnix ω to be 3 years ensures that the ratio between the number of data-points and the number of assets Q is greater than 1.5 and therefore the correlation matrix is always well behaved.Choosing the parameter ζ for the minimal variance portfolios.Figure , which maps, from a chosen set of eigenvectors, the significant ones at a time k = {0, 1}, k = {0, 1, 2, 3} and k = {0, 1, 2, 3, 4, 5} is shown. A lower state indicates that eigenvectors with smaller eigenvalues are used to construct this state.In Fig. . The biggest changes in these three cases occur in October 2005, October 2008, October 2011 and July 2015.Increasing the number of eigenvectors also increases the number of possible states but the number of actual occupied states is much lower than the number of possible states  and the connection between systemic risks and correlation matrices.42. From 2000 until 2017, there is clear jump in 2008 in all investor approximations γ∈ supporting research by Preis et al.1, who found the diversity breakdown in times of market stress.The S&P 500 analysis shows similar transition points to the pure PCA analysis of a correlation matrixtc of a financial system by applying eigenvalue decomposition on the similarity matrix ζ and determine whether t′ belongs to the same similarity subspace as t < t′. If at t′ another set of eigenvectors is needed, then t′ is a transition point tc. By increasing the number of eigenvectors and therefore gaining descriptive information of the similarity matrix, more transition points can be identified and more subtle distinctions between states can be found. These points in time highlight the investment shifts in the market, which can be small or larger structural changes as it was the case in 2008 and 2011. These critical transitions are linked to volume changes in the S&P 500.That implies that one can measure the transition point In summary, our method to find critical transition points within a financial market is based on the similarity between mean-variance portfolios at different times. The resulting matrix is analysed by eigenvalue decomposition, which uncovers the temporal development of different similarity levels. As a last step, the subspaces formed by the eigenvectors are mapped to unique states along the time axis. This method allows to find two new aspects of financial markets: Market phases must be classifiable by an investor approximation, times of systemic risk fall into transition phases. These results are complementary to the known stylized facts and should be incorporated when constructing new general financial market models.44 in its classical form. This is an optimisation problem, where one has to minimise the variance and maximise the return of a portfolio p. Moreover, by restricting the portfolio components pi to be positive for all available assets N, we only allow long positions. The resulting selection problem is solved by minimising the cost-functionμi is the expected return of asset i and C the pearson covariance matrix. The parameter γ is used to balance the trade-off between risk and return. A value of γ = 0 would result in only minimising the variance, while a value of γ = 1 would cause portfolios to be only optimized for maximum return.In order to approximate an investors risk aversion, we use the mean-variance model by Markowitzet al.46 for fitting generalised linear models in order to generate efficient portfolios for a specific γ. As proposed by Altenbuchinger et al.47, one can incorporate the equality constraint pk one obtains an extreme value for pk and ps. With the help of the second derivative and curve sketching, one can fulfill the constraint pi ≥ 0 and a coordinate descent step can be constructed. Iterating this update step in combination with active set cycling49, allows the generation of thousands of portfolios in a reasonable time frame.We implemented a coordinate descent algorithm as used by Friedman Calculate the participation τ, where τ is a given threshold. τ is set to 0.01 in this paper. Each row of S now represents a state.Map U to a state matrix S = U > s between 1 and the number of unique states found in S, where state 1 is the state combined from the eigenvectors with the lowest eigenvalues.These states are now mapped to a state number S → s is returned.The state time-series The algorithm for mapping a given set of eigenvectors to a state works in 4 steps:50.The financial time-series data for the S&P 500 was downloaded with the QUANDL50 data interface from the WIKI database.The data we used to create the mean-variance portfolios was downloaded from the WIKI Quandl database


Countries and cities are likely to enter economic activities that are related to those that are already present in them. Yet, while these path dependencies are universally acknowledged, we lack an understanding of the diversification strategies that can optimally balance the development of related and unrelated activities. Here, we develop algorithms to identify the activities that are optimal to target at each time step. We find that the strategies that minimize the total time needed to diversify an economy target highly connected activities during a narrow and specific time window. We compare the strategies suggested by our model with the strategies followed by countries in the diversification of their exports and research activities, finding that countries follow strategies that are close to the ones suggested by the model. These findings add to our understanding of economic diversification and also to our general understanding of diffusion in networks. The probability that a region will develop a particular research activity increases with the number of similar activities in neighboring regions. Here the authors analyze diffusion strategies and show that it is not only important to know which activities to target but also when to target them. In a world where the probability that a region will enter a new economic activity increases with the presence of related activities, one of the main challenges of regional economic development is to identify the activities that a region should target. Regions may want to focus on activities that leverage local knowledge, but are unlikely to open new development paths. Or they may choose riskier activities, which could be harder to develop but lead to new opportunities. In choosing the latter, there is a question on whether there is a particularly beneficial time to leap into a new territory.2, or develop a new industry4, technology5, or research activity7, increases with the number of related activities present in that location, or with the presence of that activity in neighboring locations8. These findings, which are evidence of the social, technological, geographic, and economic constraints to knowledge diffusion, have been modeled using network methods9 that allow scholars to connect related activities and by new datasets that summarize patterns of co-location1 and co-production7, as a way to measure the relatedness of activities. This literature has given rise to a nuanced understanding of the empirical path dependencies that shape economic development, but has left unanswered questions about the optimal strategies needed to traverse these development landscapes.During the last decades a growing literature has shown that the probability that a region will start exporting a new product13. Hence, it presents a good point to start thinking about strategies for cases when knowledge diffusion is constrained by the relatedness of activities. An important distinction within this networks’ literature is the difference between simple and complex contagion16. Simple contagion involves situations where transmission requires contact with a single individual17. Complex contagion involves reinforcement by multiple sources16, and hence, is a more accurate representation of development processes involving knowledge diffusion, where the probability of success is known to increase with the presence of related activities8. Therefore, we can create models that optimize knowledge diffusion by building on the idea of complex contagion16.The literature on network diffusion, on the other hand, has explored how the structure of networks, and the seeding of epidemics or information, affects diffusionHere, we formalize the problem of identifying optimal economic diversification strategies as a problem of strategic diffusion in the presence of complex contagion. We explore this problem for simple and complex network topologies, finding that strategies that minimize the total time needed to diversify an economy target highly connected nodes at an intermediate step of the process. We compare the strategies with the empirical behavior observed for countries in the diversification of their exports and research publications, finding that this empirical behavior is similar to the strategies suggested by our theory.1 and research areas are connected if the same scholars are likely to have published in both of them7 . This empirical law means that the cost of developing a related activity is much lower than the cost of developing an unrelated activity, since the probability of success is much higher for related activities. For the product space and the research space this power function has a slightly super-linear form, which is respectively: y ≈ 0.16x1.03 for the product space and y ≈ 0.74x1.09 for the research space (See SI for more information).We can model this behavior by noticing that in both cases the probability that a country will start exporting a new export, or become active in a research area, increases as a power of the fraction of related activities present in that location (aij is the adjacency matrix connecting related activities i and j (aij = 1 if there is a link and 0 otherwise), Mj is a memory vector indicating if activity j is active (Mj = 1) or inactive (Mj = 0) , ki is the total number of activities related to activity i (whether they are active or not) , α = 1 means that the probability of activating a node is linearly proportional to the number of related activities present in a location, and α > 1 means that the probability of activating a node increases concavely with the number of related activities present in a location. Both the product space, and the research space, exhibit a behavior that is slightly super linear  can be in one of three states: active (A), potentially active, (P) or inactive (I) Fig. . Potenti18, Peru19, Chile20, and Indonesia21, are just a few examples that have made increasing the diversity of their economies an explicit development goal. Certainly, one could focus on other objective functions . For now, we leave the extension to other objective functions as a future exercise and focus on minimizing the total time needed to activate all of the nodes in the network.Since developing activities is costly, we model diffusion sequentially, by choosing one potentially active node as an activation target at each time step (the target). A strategy, therefore, can be described as an ordered sequence of activation targets. Hence, the strategic problem that we need to solve is to identify a sequence of targets that minimizes the total time needed to activate all nodes in a network. Recently, increasing economic diversity has become an explicit economic development goal for multiple countries. Saudi ArabiaAlso, we do not consider strategic interactions (situations in which multiple diffusion processes are competing for the same nodes in the network). While we acknowledge that the problem of strategic interactions is an interesting one, we use instead what is a standard assumption in economic development. This is that world markets are relatively large compared to domestic markets. This assumption is validated by the data, since—on average—new entrants represent less than 1 percent of the global market of a product. So the real competition that new entrants face is not with each other, but with large incumbents. Hence, we focus on exploring solutions for the problem of strategic diffusion in the case of a single objective function (diversification) and in the absence of strategic interactions. Yet, even in this simple case, the problem of finding optimal sequences for complex contagion is computationally expensive, so we make simplifications that make the problem tractable. Going forward, we call the sequences that minimize the total time needed to activate the network an optimal diversification strategy.24  Fig. , a gener24 Fig. ). We use25. Finally, the random strategy allows us to describe what to expect in the absence of strategic choices, and hence, is an important null benchmark that any sensible strategy should beat.We benchmark our strategies using five simple diffusion strategies: random strategy: where we target potentially active nodes at random; high-degree strategy: where we target the potentially active node with the highest degree; low-degree strategy: where we target the potentially active node with the lowest degree; greedy strategy: where we target the node with the highest probability of activation; and majority strategy: where we target the potentially active node with the highest number of active neighbors . The motivation for each of the five strategies is as follows. On the one hand, we have low-hanging fruit strategies, such as greedy and low degree. These are strategies that target easy to activate nodes, but fail to consider the future strategic value of these nodes (how many nodes they will help activate in the future). On the other hand, we have more ambitious strategies, such as majority and high degree, which target nodes that could be difficult to activate early on, but that can help activate other nodes later in the process. The high-degree strategy is particularly interesting as a benchmark because it is a common heuristic in problems of simple contagionZ nodes α) does not change unless the hub is active. After the hub is active, the probability of activating peripheral nodes grows to (2/3)α.Consider a wheel network populated by des Fig. . A wheelWe start with a single active peripheral node. The greedy strategy in this case is to always develop the activity with the highest probability of success, and hence, to activate 1/3 of the peripheral nodes before targeting the hub. The majority strategy would target the hub after one peripheral node is infected, and would be almost identical to the high-degree strategy. The low-degree strategy would target the hub last.We can obtain an optimal strategy by leveraging the symmetry of the wheel network to write an equation for the average total time needed to activate the full network.L be the time when we target the hub, measured as the number of peripheral nodes that have been activated. Also, note that the expected time t required to activate a node with activation probability p is t = 1/p. Then, the total time required to activate the entire network will be equal to:α is the time required to activate each of the first L−1 nodes, α is the time required to activate all remaining peripheral nodes after activating the hub, except for the last one which takes one unit of time.Let L and α ). When α approaches zero, relatedness does not matter and therefore, activation times do not depend on when you target the hub. As α increases, and we dial in the importance of relatedness, an optimal solution emerges. For α = 1 the time is minimized when about 7% of all nodes are active. For α = 2, the optimal time to target hubs is when 10–15% of the peripheral nodes are active. This optimal window is sooner than what we would obtain from the greedy strategy. All other strategies  perform poorly. Also, we note that the optimal window is asymmetric. Since the functional form of Eq. Figure x-axis shows the fraction of the network that has been activated, and the y-axis shows the degree of the nodes that is optimal to target at that time. The optimal value (L*) can be obtained by setting dT/dL = 0:Figure α, meaning that the value of using the right strategy is small when relatedness is not important (α << 1), but large when relatedness is substantial (α ≥ 1). Finally, we find the optimal target values increase with α, meaning that we need to target hubs later in the process when the importance of relatedness is stronger.The wheel network model has four important implications. First, it tells us that the optimal window of time to target the hub activity is lower than we would expect from the greedy strategy that focuses only on the most related activity m hubs connected to kH low degree nodes that form themselves a random network with average degree kL << kH. The total number of low degree nodes is n. . For illustration purposes, we consider generalized wheel networks with 1000 nodes (10 hubs and 990 low degree nodes) although the results are not too sensitive to the number of hubs.Figure Figure 26  and the majority strategy (with probability 1−p) to minimize diffusion time. This mix of strategies beats all benchmarks, and performs relatively better when relatedness is more important (higher α). The inset of Fig. α. Figure Finally, we explore diffusion in scale free networks26 Fig. . Scale-fWe now can bring the theory to the data by looking at the average strategies followed by countries in the development of new exports and research areas. Like we did for our models, in Fig. Figure These results show that the process of diversification of countries in the product space and research space is—on average—close to what we should expect from a model of strategic network diffusion focused on minimizing the diversification time.27. While many mechanisms have been proposed in the past to explain such agglomerations, knowledge diffusion has emerged as one of the main drivers of related activities at multiple scales, from cities, to regions3, to countries1. The main finding supporting this idea is the fact that the probability of entering an activity increases with the number of related activities present in a location, or in nearby locations, probably because similar activities require similar inputs 31. Yet, despite the prevalence of this relationship, little is known about the strategies that are optimal to maximize diversification when knowledge diffusion is constrained by a network of related activities.Scholars have long observed that similar productive activities agglomerate1 and the research space7. Yet, one could extend these ideas to other objective functions, such as total contribution of exports to GDP, economic complexity32, or the carbon intensity of products23. To do this, one would assign each node to a value , and would search for a strategy that finds the sequence that maximizes an aggregation of these values. In our case, we focus on diversification as an objective function because it represents a convenient baseline, one where all nodes have the same value. Yet, other functions could be explained as well.Here, we explore strategies to maximize knowledge diffusion in situations where the probability that a region will enter an activity increases in the presence of related activities. Of course, this is not a simple problem, so we present a stylized version of it focused on maximizing diversification for a few theoretical networks and two previously published networks of relatedness: the product spaceHere, we explore this problem analytically for simple network structures, and numerically for complex networks, to show that the strategies needed to maximize diversification in the presence of related activities are dynamic, and require focusing not only on which activities to target, but on when to target highly connected activities. Our findings suggest that efforts to target highly connected activities and research areas should be optimal at an intermediate, and relatively low, level of diversification. These findings improve our understanding of strategies that can help maximize knowledge diffusion, and also, add to our general understanding of diffusion in networks.7.The details of all data and methods used are given in Supplementary Notes Supplementary Information(PDF 1515 kb)Peer Review File(PDF 308 kb)