Similar to intelligent multicellular neural networks controlling human brains, even single cells, surprisingly, are able to make intelligent decisions to classify several external stimuli or to associate them. This happens because of the fact that gene regulatory networks can perform as perceptrons, simple intelligent schemes known from studies on Artificial Intelligence. We study the role of genetic noise in intelligent decision making at the genetic level and show that noise can play a constructive role helping cells to make a proper decision. We show this using the example of a simple genetic classifier able to classify two external stimuli. Cellular decision making and the determination of cellular fate is performed by superimposing the pattern of extracellular signalling with the intracellular state given by locations and concentrations of chemicals within the cell. This process is regulated by the program encoded in the cellular genome. Understanding the principles of this programming and, especially, how this program is executed is a key problem of modern biology, for control of this program would provide insight into new biotechnologies and medical treatments.Our brains is able to make intelligent decisions because they are controlled by neural networks, i.e., networks based on the communication between huge amount of cells. But can intelligent solutions be executed at the level of genome? First, we should define what we will understand under intelligence. For this we will use some basic schemes developed in the field of Artificial Intelligence. One of the basic schemes of intelligence has been suggested by Frank Rosenblatt, who has called such systems as “perceptrons”. Basic perceptron was able to classify several external stimuli and provide binary output. Could such an intelligent decision making be observed in the performance of gene regulatory networks? It was long known that cells can adapt to and anticipate the stress but it remains not completely clear whether this is a result of intelligent learning or something else. For example, in 2008 Saigusa et al.  have shoOn the other hand, recently it has been demonstrated that gene expression is a very noisy process . Both inS1 and S2 be linearly separable sets in n-dimensional space with a separation threshold T, i.e.,T is linked to a current cellular state, such a scheme can learn the classification rule as an intelligent perceptron.To show this we have designed a simple genetic network able to classify two external stimuli. The form in which our intelligence will take in this paper will be in the ability to perform linear classification tasks. Linear classification describes the ability to successfully discriminate between two sets which are linearly separable. Let suicide’ . Additiox ∈ :x =  represents the n-dimensional input and the weight vector w =  and the value T describe the separation hyperplane. As can clearly be seen from  denote the ith nodal mRNA concentration and p denote the ith nodal protein concentration the rate equations are as follows:Letting β is some positive constant  and f(x) is a sigmoid function switching from 0 to 1 if x is increased. The matrix A represents the topology of the genetic network. Aij > 0 implies that the ith protein is an activating promoter for the jth gene, Aij < 0 implies that the ith protein is an inhibitor for the jth gene and Aij = 0 implies that there is no transcriptional relationship between the ith protein and jth gene. In addition to this nodes 3 and 4 are self promoting, although this is simply to increase the speed of convergence and will not qualitatively affect the dynamics. For example the A matrix for the network in m and p are in  they will never leave this range. As a generalised chemical concentration could not be so rigorously contained we should consider these as concentrations relative to some maximal concentration level.Where w1 and w2 and thus the weightings for linear classification. If the weights have been trained as a result of the evolution, then not only w1 and w2, but also other connections could have a different value obtained as a result of training. It is possible also that the inputs will activate node 3 and repress node 4 with a different rate. This will lead to some changes of the separation line shown in the w1 and w2 will be adjusted.As discussed earlier we wish to have some mechanism by which the weightings  can be adjusted. As this would represent a multiplicational transcriptional strength we consider a mechanism of phosphorylation. We now require that the proteins produced at nodes 1 & 2 must be activated via phosphorylation by some external kinase concentration. In this way we can consider the rate of production of phosphorylated protein as the product of concentrations of the unphosphorylated protein and the phosphorylating kinase concentration. It is by this mechanism that we justify the ability to adjust the values of In From here onwards we will be interested in noisy systems where noise is an extrinsic noise added to inputs. Hence, rather than using techniques of stochastic simulation popular in Mathematical Biology we will be using deterministic systems adjusted by the addition of some stochastic term to inputs. Effect of this can clearly be seen in For spoiled perceptron, however, adding noise can surprisingly restore a correct classification manifesting the effect of stochastic resonance. The term stochastic resonance is somewhat of a misnomer as in this case we make no reference to acoustic or vibrational resonance. Instead what we mean is that some measure of functionality of our system is improved by the addition of some optimal intensity of noise. This optimal noise should be some finite value of the variance of an additive normally distributed noise term. This implies that our system should perform worse after a certain amount of noise. This makes sense intuitively as an infinitely noisy system would not be able to facilitate any order.N input sets ith input set, and letThe inputs to our system are generated as a pair of random numbers uniformly distributed between zero and one and then classified according to the unspoiled threshold. Next, the same inputs have the Gaussian noise added to them and are classified again this time using the spoiled threshold. The two classifications are then compared and compiled into the accuracy score as follows. Considering a set of If we perform this accuracy analysis with a spoiled threshold we observe The governing equations for our genetic network model were simulated using a 4th order Runge-Kutta numerical integration scheme. Normally distributed random variables were generated using the Box-Muller transformation for uniform random variables. When considering accuracy we calculated it in the following way, using a Monte Carlo style approach. We generated a set of 10000 pairs of random variables, distributed randomly on the interval . These pairs became the initial concentrations of nodes 1 and 2 respectively. The network is then simulated for these initial conditions and after a sufficient amount of time the concentration of node 3 will have converged on either 1 or 0. This value is taken as the output for that simulation. We then repeat the simulation for the same pairs of inputs but this time with a normally distributed random variable added to each of them and the initial concentration of node 0 adjusted. We compare the results of this new simulation with the simulation of the same pair of input concentrations.n inputs  and measure the accuracy of the output assignment functionality of the algorithm both with and without noise. The noise is represented by a random variable from a Gaussian distribution with mean μ = 0 and variance σ2 = z where z will be increased from 0 to 1 to represent our increasing intensity of noise. Our accuracy measure simply gives the success rate with which the algorithm correctly classified all members of the test set when given the new, spoiled system to work with. In practice we calculate this using a Monte-Carlo style approach, a set of N (10000 in practice) input vectors are generated, each consisting of n uniformly distributed random entries from the range . We classify this set according to the unspoiled threshold T1 to getT2 to giveHere we will spoil the threshold value ed as in . Accuracw1 = w2. Our input values x1 and x2 are uniformly distributed random variables between 0 and 1, and as such will each have a probability distribution function (pdf) given by:X = x1w1+x2w2, thenNow that we have demonstrated the existence of this effect of stochastic resonance it would be prudent to attempt to explain it analytically. In doing so we must think more rigourously about our random variables and their distribution functions. For simplicity let us first consider the 2-input system with μ = 0 but with changing variance σ2. The normal distribution has the following pdf:Now we must consider the addition of noise. Our noise is distributed normally, always with X. The sum of two independent normal variables is normally distributed itself with μ = μ1+μ2 and σ by w.We wish to combine 2 lots of noise (1 for each input) and as in this case both of our weights are the same we can simply combine our noise terms together and add them to This gives:T1 be the pre-spoiling threshold and T2 be the post-spoiling threshold. For a point x this probability is given by:Now we must consider how to formulate the system as a whole. We will consider the concept of accuracy as before, where for each point which is successfully allocated before spoiling, we consider the probability of it still being successfully allocated after spoiling and noise. Now let x < T1x > T1If x, remembering to use our probability density function f(x). As the distribution of x is independent to the distribution of the additive noise we can simply multiply the probabilities:Now we must apply this to the entire distribution of f(x) represents a probability distribution function we know that its integral over its whole range  must be equal to 1. Using this we can simplify further:This expression containing double integrals can be tidied up in terms of the error function:σ2 and comparing it with the simulated results using the same coefficients we can see perfect correlation . First consider our input variable X ∼ w1X1+w2X2+…+wnXn. Each wiXi is a uniformly distributed random variable on the range . We can generate the probability distribution function for a distribution such as this from the following function [To generalise this model we must consider a system with function .fn(s)=1Where:n = 2.Theoretically it is now possible to produce the probability distribution function for any number of independent inputs but due to the nature of this equation it is difficult to treat this general formula analytically. We will proceed by generating it for the case n = 2 we arrive at the probability distribution function for 2 independently weighted inputs. Labelling such that w1 < w2If we evaluate this for n independently weighted noise terms is simpler. We can simply consider a single Gaussian distribution with a modified variance Considering the addition of LetThenAndFrom this we can see how to create our new variance.σi will be equal so we have:n input system to a single variable distributed according to fn(s) and a single noise term, defined as above.But all σ as stated and defining W as the sum of all weights we end up with a similar description for the full system as before.n inputs with independent weightings.Defining σ we take T2 to be the independent variable and σ to be a parameter.Earlier we discussed the idea of threshold spoiling acting as some sort of mechanism for providing robustness to noise. Now with this in mind we can refer back to  but rathT2 is 1 and equal to T1 as would be expected. Increasing the noise intensity also increases the optimal T2  to be adjusted within the cellular life rather than having to be adjusted evolutionarily. Considering this system in a presence of noise we observed, demonstrated and quantified the effect of stochastic resonance in linear classification systems with input noise and threshold spoiling. To explain a mechanism of this effect we have considered a simple classifying perceptron and have shown that analytical results match the numerical simulations. The consideration of linear separation in noisy environments is relevant due to the inherantly noisy nature of gene expression in cells both due to extrinsic or intrinsic factors.σ2 > 0.1 spoiled perceptron has better accuracy in classification than not spoiled one. It could lead to speculations that, since some noise is certainly present in gene expression, classifying genetic networks will evolve towards shifting the classification threshold to compensate an effect of noise.Whereas an idea to spoil the threshold of classification may look a bit artificially in a sense that we first corrupt a classification, and then restore it, the w = {w1, …, wn} denote the desired weight set and lth iterative attempt at learning the desired weights. Also let xj is a set of input values for which we have the desired output given by O(xj). Then we update our weights as follows,The design presented allows for the ‘contextualisation’ of classification whereby the same network implemented in a different scenario could perform an entirely different separation task depending on the stimuli present. While this allows for a great deal of flexibility we would like to explore the idea of the implementation of some kind of perceptron learning in the same vein as the delta learning rule, a simple version of back propagation learning. In this learning method weights are updated by a process of comparison of the perceptron output to some ideal result for the given inputs. Mathematically we consider it as follow. Let While this is more computationally intensive than simply setting our weights as desired as in the current implementation it has a major advantage in the sense that it is results driven. With this kind of learning we are guarenteed for our system to conform to our desired outputs as provided in the test sets. When setting weights directly we are not guarenteed that we will get the results we desire, simply that we will have the weights given.The thinking behind backpropagation learning involves the system performing a comparison between the result which it has arrived at and some ideal result. By considering discrepancies between these two results it can then make adjustments to the process by which it arrived at its result in order to move closer to the ideal result.j of some training set O, where we have a set of inputs along with their desired output, we should compute the following:This kind of learning requires a large degree of reflexivity on the behalf of the network and as such it is difficult to imagine a biological implementation of such a learning technique. Despite this we feel it is worth considering whilst on the topic of perceptrons as it is an extremely effective technique. The procedure is simple: for each element α is some sort of learning rate, if we choose α too large then we may end overcorrect and adjusting the weights too far in each update, but the smaller it is chosen then the longer it will take to arrive at the desired weights.Here wi is performed for each i and looped through for each entry in the training set. The whole process is then repeated until the error in the system is sufficiently small or a maximum number of iterations is reached. For a training set of N input/outputs we have:α = 0.01 was found to be suitable for optimal learning speeds and for what follows, the unspoiled threshold has T = 1.This improvement to T in F(X) as defined in the previous section and adding the noise to each x1 and x2 from the test set before it is fed into F(x). The accuracy is measured by considering the error from the actual intended weights. This is done by finding the Euclidean distance of the algorithms attempts at the weights to the correct values:Previously, we have just been examining the output assignment functionality of the algorithm but the ideas of stochastic resonance could also be applied to the learning algorithm. As in the previous case we must spoil the system in some way and then we will examine some kind of measure of accuracy for increasing intensity of noise. The way in which this will be implemented is by spoiling the value of wnew1,  and wnew2, . As is clear from σ2 ≈ 0.12) rather than a maximized accuracy rate as before.The algorithm was allowed to run through the test set 2000 times before arriving at its final attempt at the correct weights Another simulation featuring independently changed weights shows that the resonance still applies in this case. It would be of interest to perform a more thorough investigation into this effect and to try and describe it analytically if possible. A better understanding of this effect and the circumstances in which we can find it would be of great interest for applications in learning algorithms in general. Of additional interest would be to investigate the possibility of a genetic implementation of learning of this kind. It is not clear whether such an implementation would be possible, if it were possible it would undoubtably be much more complex than our linear classification network. If possible it’s construction and simulation would certainly be of great interest.Finally, construction of intelligent intracellular gene regulating networks is the hot topic of synthetic biology, e.g., see for a review , and her


A major desire of individuals with spinal cord injury (SCI) is the ability to maintain a stable trunk while in a seated position. Such stability is invaluable during many activities of daily living (ADL) such as regular work in the home and office environments, wheelchair propulsion and driving a vehicle. Functional neuromuscular stimulation (FNS) has the ability to restore function to paralyzed muscles by application of measured low-level currents to the nerves serving those muscles.A feedback control system for maintaining seated balance under external perturbations was designed and tested in individuals with thoracic and cervical level spinal cord injuries. The control system relied on a signal related to the tilt of the trunk from the vertical position (which varied between 1.0 ≡ erect posture and 0.0 ≡ most forward flexed posture) derived from a sensor fixed to the sternum to activate the user’s own hip and trunk extensor muscles via an implanted neuroprosthesis. A proportional-derivative controller modulated stimulation between trunk tilt values indicating deviation from the erect posture and maximum desired forward flexion. Tests were carried out with external perturbation forces set at 35%, 40% and 45% body-weight (BW) and maximal forward trunk tilt flexion thresholds set at 0.85, 0.75 and 0.70.Preliminary tests in a case series of five subjects show that the controller could maintain trunk stability in the sagittal plane for perturbations up to 45% of body weight and for flexion thresholds as low as 0.7. The mean settling time varied across subjects from 0.5(±0.4) and 2.0 (±1.1) seconds. Mean response time of the feedback control system varied from 393(±38) ms and 536(±84) ms across the cohort.The results show the high potential for robust control of seated balance against nominal perturbations in individuals with spinal cord injury and indicates that trunk control with FNS is a promising intervention for individuals with SCI. A major desire of individuals with spinal cord injury (SCI) is the ability to maintain a stable trunk while in a seated position . Such stPreliminary steps in designing advanced control systems for maintaining trunk posture involve studies to assess the strategy used by the intact central nervous system to mediate trunk balance in individuals with no known neurological disorders. These studies include both experimental observations of the static and dynamic behavior of trunk posture in a seated pose –7, as weRecent studies have shown that a self-righting control system could be deployed to automatically return the trunk to an erect posture from forward-flexed positions by using a measure of trunk tilt as a feedback signal to modulate stimulation to the trunk and hip extensor muscles appropriately –13. The Five individuals with paraplegia at different thoracic and cervical levels participated in the experiments. Table The volunteers sat on a chair which was placed in the work volume of a 16-camera Vicon motion capture system. A schematic of the setup is depicted in Figure Retro-reflective markers were placed at the left and right posterior-superior iliac spines, shoulders and on the C7 cervical spine. Others were attached to the four corners of the biomechanics force platform  on which the subject sat. With these markers, it was possible to compute the total trunk tilt angle in the sagittal plane. This served as a confirmatory signal for the trunk tilt sensor output and also for measuring response time of the control system. A step signal from the real-time xPC Target computer synchronized the beginning and end of a trial capture with the Vicon computer.A linear actuator  capable of applying forward pull forces to the trunk via a rope attached to a belt wrapped around the chest was affixed to the laboratory wall in front of the volunteer. The pull force was programmed accurately as a function of the volunteer’s body weight (BW). Depending on the ability of the volunteer to tolerate the pulls, several load levels were tested  starting from the lowest to the highest that could be tolerated without failure of the controller to return them to erect. Although experiments were conducted for 7 force pull levels it was observed that for most of the subjects the smaller values (20% to 30% BW) were too small to cause significant movement away from vertical while the 50% BW setting caused most of them to fall too far for the controller to act on time to restore balance. Hence subsequent details were reported only for the 35%, 40% and 45% BW magnitudes. The force pulse with respect to time was a trapezoidal shape that ramped up in 100 ms, dwelled for 100 ms and ramped down in 100 ms. The magnitude of the force output from the actuator was set in a block of the Simulink control software. The value was used by the xPC-Target computer to set the current (in mA) to the actuator amplifier.Prior to each experiment the volunteer sat erect with their trunk supported by an adjustable back-rest of the seated platform. The purpose of the backrest was to prevent the volunteers from falling backward as the controller did not stimulate hip or trunk flexor muscles to control trunk extension.In this study the main input variable to the muscles was the stimulation pulse-width (PW). As shown in Figure During steady state sitting, all the muscles were excited at their baseline values – being the muscle PWs suitable to keep the trunk at the erect posture. The PWs for some or all of the muscles were varied (from baseline) to oppose changes in trunk tilt as measured by the accelerometer sensor worn by the subject. Table k was defined as [Initial testing with a proportional controller proved that the stimuli did not recruit muscle force fast enough to generate the corrective moment required to resist the perturbations. A derivative term was added to improve controller performance. It was found necessary to filter the derivative term as any noise in the sensor signal was amplified and resulted in rapidly varying muscle responses that were uncomfortable for most of the subjects. The resulting generic form of the controller equation in digitized form at time tfined as :1C(tk) was the controller term, D was the derivative term KP, and KD were the proportional and derivative gains respectively and TD was the derivative time constant (secs) defined as TD = KD/KP; N was the derivative filter constant (2 ≤ N ≤ 20) and Δt was the sampling time (secs). A normalized form of the error signal e was computed using the equation:The first term in equation  was the amin was the upright threshold setting , a was the current accelerometer reading, , a was tFor FNS application, the output from the controller as defined by equation  passed oamin; amax. The baseline PWs were earlier determined as the optimum settings that allow for a stable erect seated posture. The saturation PWs were those found to be the maximum above which no additional muscle force could be elicited from the muscle.In equation ,  was thThe controller flow diagram is shown in Figure For each subject 18 trials were analyzed. The combination of three perturbation amplitudes  and 3 flexion threshold levels  led to 9 conditions. An open source random allocation software  was usedSubjects were unable to tolerate the force pulse perturbations without the controller active. The applied disturbances, even at the lower level of 35%BW, resulted in losses of balance and rapid forward flexion of the trunk that risked potentially injurious falls from the chair. For safety reasons, it was therefore impossible to compare the controller to a baseline consisting of stimulation suitable for quiet sitting.Settling TimeTwo global performance metrics were calculated to quantify the operational effectiveness of the control system for each subject: a) settling time and b) response time. Both can be deduced from the data in the experiments in this study.b)Response TimeThe settling time was defined as the time elapsed from the application of an ideal instantaneous step input to the time at which the output (trunk angle) has returned and remained within a specified set band. In this study, we chose to define it as the time from the application of the pulse perturbation input to the time the trunk angle returned and remained within 5% of its original steady-state value prior to the perturbation.The response time was defined as the time elapsed from the onset of the stimulus change from the controller to the time when the trunk angle reached 10% of its peak amplitude following the perturbation.The results reported here were the averaged values over 6 cycles . Figure Figures Figure Figure The mean settling times for the 5 subjects are shown in the bar-graphs in Figure The mean response times for the 5 subjects are shown in the bar-graph in Figure We have developed a feedback control system for control of seated balance in the sagittal plane after spinal cord injury and tested the control system in 5 individuals: 3 with thoracic and 2 with cervical level injuries. The feedback control system used the component of acceleration due to gravity to measure the overall tilt of the trunk. The control system worked as expected to restore seated balance in all 5 subjects tested with perturbation amplitudes up to 45% BW and with flexion thresholds as low as 0.7 as illustrated in Figures In the work reported in –10, a siAnother recent attempt to use feedback control for seated balance was that reported by Vanoncini et al. . Our appWhereas in  best resVanoncini et al.  set a maThe closest concept to system response time was that of electromechanical delay (EMD) , 18. TheEven though the results reported here indicated encouraging outcomes for control of seated balance after SCI, there were a number of limitations that restrict the applicability of the results in a way. First, the muscle recruitment curves were assumed to be linear. In general most muscle elements exhibit nonlinear force-PW characteristics . The effThese initial results support the feasibility of automatically controlling seated balance with FNS. Perturbations up to 45% BW were successfully rejected by the controller. This implied that such a controller would be suitable for maintaining a stable trunk during activities of daily living and prevent falls during wheelchair propulsion or when driving a vehicle without the need for restraining devices that would otherwise decrease the workspace around the user. Further studies should include control in the coronal plane as well and also ability to set and attain any desired posture other than the erect posture within the typical work volume of a seated operator.


Trypanosoma cruzi-infected individuals have been conducted. Text message interventions, a subset of mHealth, has shown to improve appointment attendance and is relatively simple to set up, thus making it an ideal mechanism to facilitate communication with individuals in low-resource settings.Diagnosis of congenital Chagas disease occurs at 9 months of age, making effective treatment challenging due to loss to follow-up. Mobile health  has been utilized to improve communication and treatment adherence in many chronic diseases, although no studies of mHealth in The aim of this study is to understand the acceptability, utilization, and barriers of an SMS-based appointment reminder to confirm a post-partum home visit to women in Tucumán, Argentina and whether these factors differ in urban and rural populations.Women that tested positive for Chagas disease were invited to receive SMS reminders of their follow-up 4-week postpartum home visit. Demographic information and SMS contact preferences were collected at hospital discharge, and variables on mHealth utilization and barriers were recorded at follow-up.77 (70.6 %) of women possessed a cell phone for personal use. All eligible women owned phones compatible with SMS messages. The appointment reminder SMS was widely accepted with 64/72 (88.9 %) enrolled women receiving the SMS message and 58/64 (90.6 %) replying. Ninety-two percent of women stated that the text message was a useful reminder for the follow-up home visit. Women living in rural areas were less likely to own a cell phone for personal use and were significantly less likely to have internet access on their phone than women living in urban areas . Furthermore, women from rural areas faced barriers to mHealth uptake such as change of phone number and response to messages from the hospital team at higher rates than women from urban areas, although these differences were not statistically significant.There is generally widespread acceptance and utilization of mHealth among this group of women with access to cell phones. However, there are still many barriers to overcome before mHealth interventions attain complete penetration in a population, most notably the issue of cell phone for personal use. Trypanosoma cruzi. It is a major cause of morbidity and mortality in the Americas due to the fact that pregnant women with Chagas disease can transmit T. cruzi to their fetuses [Chagas disease, or American trypanosomiasis, is caused by the protozoan parasite  fetuses –3. The d fetuses , 5. It h fetuses , 6–8. CoText message interventions (TMI) are an important part of the larger strategy of mobile health , which entails the utilization of mobile technology in healthcare. Almost 85 % of residents of developing countries report cell phone ownership , making Our previous research has shown the interest of pregnant women in Argentina to receive text messaging as a form of communication with healthcare staff . PreviouThe aim of this study is to understand the acceptability, utilization, and barriers of a SMS-based appointment reminder to confirm a post-partum home visit in Tucumán, Argentina. We also sought to elucidate whether there is a difference in these factors in urban and rural populations.T. cruzi transmission in a cohort of seropositive women [This prospective observational study was carried out from December 2011 to April 2013 at the Instituto Maternidad Provincial Nuestra Señora de las Mercedes, a public maternity hospital located in the city of San Miguel de Tucumán, the capital city of Tucumán province. The maternity is the referral ward of northwest Argentina and has around 9000 deliveries per year. This study was nested within a study designed to detect the rate of congenital ve women .The inclusion criteria for the main study included all consecutive live births occurring from April 2011 to April 2013, irrespective of the gestational age or route of delivery . Only those women who tested positive for Chagas disease, had just delivered a live birth, were 18 years of age or older, and residing in the province of Tucumán area were invited to participate to the main study. The province has urban and rural areas. All women invited to the study were literate as they were all able to read and sign the informed consent.be enrolled in the main studyown a mobile phone for personal use,own a mobile phone compatible with SMS technology.The specific inclusion criteria for this nested study required participants to:All eligible women were approached in person by data collection personnel and invited to participate in this study before being discharged from the hospital. Women were told that the purpose of the study was to evaluate the feasibility of using an SMS message as a reminder of their home visit, and if they accepted to participate they would be contacted by a text message to confirm the main study appointment at 4 weeks after delivery. They were also informed that they would receive credit to answer the text message. Those accepting to participate were asked to sign the informed consent in Spanish, which was approved by the Institutional Review Board of Tulane University and the Ethics Committee of the Center of Medical Education and Clinical Investigations in Buenos Aires, Argentina.The study consisted of three phases, admission to the study before hospital discharge; visit reminder and a home visit at four-weeks after delivery. The admission form was administered face to face while women were still in the hospital. This form was used to evaluate participant´s date and time contact preferences and how many days in advance they would prefer to receive the reminder. Participants were also asked whether they lived in an urban or rural area. At this time, the hospital contact number was added to the participants phone address book to ensure the number was easily identified with the study and to avoid scams. The phone compatibility was checked sending a test text message from the study phone to the phone number provided by the woman during admission process.According to the information on the admission form, the data collection team sent an SMS reminder to each participant in order to confirm the appointment for the 4-week home visit. Before sending the message the team transferred credit to each phone number so women could reply without having any extra cost. Information related to date and time of the reminder message as well as all replies from participants were transferred to a designed form in order to measure the outcome of interest.Four weeks after delivery, the data collection team performed the scheduled home visits and completed another questionnaire in person to assess if the message had been received and the acceptability of the SMS appointment reminder. The questionnaire applied at 4-week follow up was a standardized questionnaire designed to record the reception of the message, the woman’s reply to the reminder message and her impression of whether the message was useful and whether the reception time was adequate. Reasons for not replying to the message and whether she had changed phone number was the same were also collected. A second part of the questionnaire assessed the general use of mobile phones such as time she had been using a mobile phone, frequency of changing the number, payment method, main use, internet access and if they used computers they were asked place of access, internet access, frequency use and main use. This second questionnaire had the aim of obtaining information to design future interventions. This questionnaire was adapted from our previous study . The teaThe hospital team received training on interviewing techniques and information regarding mHealth and text messages. The completed questionnaires were stored at the hospital until collected by research staff.p value of less than 0.05 was considered statistically significant.Data were entered into Open Clinica Version 3.1.1-Community and analyzed using SPSS version 21. The study population’s characteristics, access to cell phones and SMS message’s responses were reported as proportions. We used a Chi-squared test to assess the relationship between living in an urban or rural area and having a mobile phone for personal use. To study differences in access to and usage of mHealth between rural and urban populations we calculated the relative risk with a 95 % confidence interval. A A total of 116 post-partum women meeting the inclusion criteria for the main Chagas study were identified during the recruitment period of the nested study and 109 were contacted during their postpartum stay and invited to participate in the nested mHealth study  stating that they always have service in their home and only three respondents stating that they sometimes have cell service at home  replies were received Fig. . Two womThe study team completed 72 four-week follow-up home visits. A total of 66 (91.7 %) women were using the same phone number registered at admission  reported that the reminder message arrived at a convenient time and an appropriate number of days in advance. Importantly, 59 (92.2 %) women stated that the message was useful, with 38 (59.4 %) women reporting that they would have forgotten about the appointment if not for the message.We also surveyed cell phone habits and practices of study subjects at the 4-week follow-up visit Table . Length Upon questioning participants about the primary use of their phones, 20 (27.8 %) reported using them primarily for texting, while only 3 (4.2 %) reported using them primarily for making calls, and 44 (61.1 %) reported using their phone for both calls and texting. Additionally, we probed participants on their access to the internet and found that 24 (34.3 %) accessed the internet on their phones and 12 (17.4 %) reported having regular access to the internet on the computer .There was a statistically significant difference in the use of internet on the subjects’ phones between urban and rural groups, with 50.0 % of urban participants reporting the use of internet on their phones compared to only 15.0 % of rural participants . Other variables that were not statistically significant between urban and rural populations but that trended towards lower values in rural populations were the use of the same phone number at enrollment and follow-up , change of phone number in the previous year  and response to the message from the monitoringteam (RR 0.87, 95 % CI 0.70–1.09; p = 0.24, Table T. cruzi infection. We found that approximately 70 % of these women possessed a cell phone for personal use and were eligible for the study. All eligible women in the study owned phones that were compatible with SMS messages and had a cell phone signal in their home at least some of the time. The appointment reminder SMS reached approximately 90 % of those individuals that participated in this study. Importantly, the SMS reminder was widely accepted and considered a useful reminder of the follow-up home visit that is crucial for the successful treatment of their children. The two primary barriers to the utilization of this program were the ownership of a mobile phone that prevented women from enrolling this study and a changed mobile phone number that compromised their follow up. Although this study did not have the statistical power to detect a difference, in general, the prevalence of barriers to mHealth utilization was higher in rural than urban populations.In this study we assessed the feasibility of using a SMS reminder in a group of post-partum women in Argentina who tested positive for Most women had the same preference to receive messages 1 or 2 days before the appointment and most had no preference on the reception time making feasible to scale up this intervention.Although most women had a cell phone for personal use, the percentage was lower than the near universal use that we had previously assessed in pregnant women in other areas of Argentina . AdditioIn addition, changes of phone number are a common occurrence and would pose a challenge with any long-term mHealth study relying on the use of the patients’ phones due to the fact that approximately 8 % of participants changed their phone number in only 4 weeks between enrollment and follow-up. Moreover, 65 % of participants reported changing their number in the previous 12 months. Thus, changes in participant phone number represent a significant barrier to the effective implementation of an SMS reminder program in studies with a prolonged follow-up period. We did not investigate the causes of the frequent changes of phone number, but one potential reason would be the frequency offers of low cost cellular service for new accounts which influences individuals to change their service provider and phone number.There is also a trend of increased barriers to the access to or utilization of mHealth in rural populations; compared to women from urban areas, women from rural areas were less likely to possess a cell phone, less likely to respond to messages from our monitoring team, and more likely to have changed their phone number, either between enrollment and follow-up or as self-reported in the previous 12 months. These data suggest that while urban and rural groups face similar barriers in their access to mHealth, rural populations encounter these barriers more frequently. This is particularly important because one of the often-stated benefits of mHealth is the potential to extend health care and health information to geographically distant or isolated locations –24. ChalT. cruzi, we only enrolled post-partum women with confirmed T. cruzi infection. This limits the replicability of the study for its general use in other population. Fewer women owned a telephone and more women lived in rural areas when comparing this Chagas positive population with our previous study of non-positive population [The strengths of this study include the in-depth questionnaires at enrollment and follow-up to assess feasibility and acceptability of a 2-way text message system. Additionally, subject location data allowed us to compare barriers to mHealth utilization in urban and rural populations. These findings can be generalized to Chagas positive population in Argentina as the population who participated in this study is comparable to the population attended in public hospitals in Argentina. However, because this was a study nested in a main study on congenital transmission of pulation . As a reImplementation of any mHealth project faces certain challenges regardless of the setting. These include adjusting to evolving technology, the selection of a mobile phone and platform that will be utilized and accepted by the study population, and the specifics of the study population, i.e. poor eyesight, impaired motor skills, etc. Implementing an mHealth study in a rural area and/or resource-limited setting necessitates the adaptation to these and other challenges, such as the availability and stability of a cellular network and electrical grid. Additional considerations include the availability of personal cellular devices, privacy and security of information, sufficient mobile credit to make calls and send/receive messages, and the frequency with which users change phone numbers. Our study has begun to elucidate the prevalence of these barriers in urban and rural populations in Tucumán, Argentina. Future studies will consider these barriers when designing and implementing an effective mHealth intervention utilizing SMS communication or a mobile internet connection.T. cruzi to test for congenital transmission, and also when necessary, its efficacy as reminder of appropriate treatment.There was a widespread acceptance and utilization of mHealth among this group of women with access to cell phones. However, there are still many barriers to overcome before mHealth interventions attain complete penetration in a population, most notably the issue of individual cell phone ownership. Future studies should investigate the efficacy of text message reminders on follow-up visits for infants born to mothers infected with


In nonlinear dynamics, basins of attraction link a given set of initial conditions to its corresponding final states. This notion appears in a broad range of applications where several outcomes are possible, which is a common situation in neuroscience, economy, astronomy, ecology and many other disciplines. Depending on the nature of the basins, prediction can be difficult even in systems that evolve under deterministic rules. From this respect, a proper classification of this unpredictability is clearly required. To address this issue, we introduce the basin entropy, a measure to quantify this uncertainty. Its application is illustrated with several paradigmatic examples that allow us to identify the ingredients that hinder the prediction of the final state. The basin entropy provides an efficient method to probe the behavior of a system when different parameters are varied. Additionally, we provide a sufficient condition for the existence of fractal basin boundaries: when the basin entropy of the boundaries is larger than log2, the basin is fractal. Dynamical systems describe magnitudes evolving in time according to deterministic rules. These magnitudes evolve in time towards some asymptotic behavior depending on the initial conditions and on the specific choice of parameters. If a given dynamical system possesses only one attractor in a certain region of phase space, then for any initial condition its final destination is clearly determined. However, dynamical systems often present several attractors and, in these cases of multistability, elucidating which orbits tend to which attractor becomes a fundamental question.A basin of attractionThe previous discussion applies typically to dissipative dynamical systems. However, for open Hamiltonian systems, where the concept of attractors or basins of attraction is meaningless, we can still define escape basins in an analogous way to the basins of attraction in a dissipative system. An escape basin, or exit basin, is the set of initial conditions that escapes through a certain exit. The Hénon-Heiles Hamiltonian is a well-known model for an axisymmetrical galaxy and it has been used as a paradigm in Hamiltonian nonlinear dynamics. It is a two-dimensional time-independent dynamical system, where orbits having an energy above the critical one can escape through one of the three different exits. It is widely known that when two or more escapes are possible in Hamiltonian systems, fractal boundaries typically appearE above the critical energy that separates bounded motions from unbounded motions. Most initial conditions leave the region through one of the three different exits to infinity for any E above this critical energy. The colors represent points that taken as initial conditions leave the region through a specific exit. With this in mind, we may intuitively understand that it is harder to predict in advance which will be the final destination of an orbit in In order to give an intuitive picture of our problem we may look at The problem is that even though, we can have an intuitive notion that This is precisely the idea of uncertainty or unpredictability which we are considering here. This remark is important since we are aware that these terms are polysemic and consequently its use in the literature might be confusing. In this paper we refer to unpredictability or uncertainty as the difficulty in the determination of the final state of a system, that is, to which attractor the initial conditions will tend to. Note that we speak about attractors for simplicity, though the discussion is identical for open Hamiltonian systems, where there are no attractors. This notion of unpredictability strongly differs from others used in nonlinear dynamics, like the Kolmogorov-Sinai entropy4The concept of basin of attraction is broadly used in all branches of science. The flow of water close to an obstacle can be described by means of basins of attraction, and their complicated structure explains the heterogeneity of phytoplankton and the information integration of the early macromolecules evolutionMany authors describe fractal basin boundaries and when discussing its associated unpredictability, some vague affirmations are found due to a lack of an appropriated measure. In particular, this has been the case with the Wada basins which have received much attention in the past few years because they are said to be even more unpredictable than fractal basins3111213basin entropy. The main idea is to build a grid in a given region of phase space, so that through this discretization a partition of the phase space is obtained where each element can be considered as a random variable with the attractors as possible outcomes. Applying the Gibbs entropy definition to that set results in a quantitative measure of the unpredictability associated to the basins. The discretization that we are considering arises naturally both in experiments and in numerical simulations. First, the experimental determination of initial conditions in phase space is physically impossible due to the intrinsic errors of the measurements. In the case of numerical experiments, the limitations of the computing resources constrain the resolution of the phase space under analysis. This unavoidable scaling error can induce wrong predictions even in deterministic models. Then, a natural question arises: how does the uncertainty in the initial condition affect the final state prediction?Our paper constitutes an attempt to give a quantitative answer to the question of the uncertainty of the basins and this is precisely the problem that we discuss here. We propose a natural way to characterize the uncertainty of the basins by defining a quantitative measure that we call et al.α called uncertainty exponent, which is the dimension of the phase space D minus the capacity dimension d of the boundary that separates both basinsA first approach to study the final state uncertainty has been investigated by Grebogi α = 1 for basins with smooth boundaries, and α < 1 for basins with fractal boundaries. The closer α gets to zero the more difficult it becomes to predict the system. If smooth and fractal basins are mixed, the uncertainty exponent can still be calculated for each boundary, although the procedure is cumbersomeThe uncertainty exponent takes the value Another approach to measure the unpredictability consists of evaluating the volume of each basin of attraction in a certain region of phase space. The ratio of the volume occupied by a single basin to the total volume defines the basin stabilityNA attractors for a choice of parameters in a certain region Ω of the phase space. We discretize Ω via a finite number of boxes covering it. Here we study two-dimensional phase spaces, so that we cover Ω with a grid of boxes of linear size ε. Now we build an application color. Each box contains in principle infinitely many trajectories, each one leading to a color labeled from 1 to NA. In practice we can use only a finite number of trajectories per box. Indeed, it would correspond to the number of times an experiment is repeated, or the number of trajectories computed in a numerical simulation. In this work, we use square boxes with twenty-five trajectories per box (if not otherwise stated) in our numerical simulations. We have seen that twenty-five trajectories per box allows fast computation and provides accurate values of the basin entropy in all the cases studied here  to log NA (completely randomized basins with NA equiprobable attractors). This latter upper value is in practice seldom realized even for extremely chaotic systems. The basin entropy in general decreases with the scaling box size ε, as explained hereafter. We now have a tool to quantitatively compare different basins of attraction.which we call ε, given a fixed ε the value of the basin entropy converges as the number of trajectories inside a box increases.Despite the fact that the basin entropy depends on the scaling box size Si = log(mi), where mi are the different colors inside a box (the accessible microstates of the Boltzmann entropy). Then the equiprobable total entropy becomes Nk equal boxes (in the sense that they have the same number of colors), where At this point, we can delve deeper into the consequences of this definition by considering a simple hypothesis, which is to assume that the colors inside a box are equiprobable, thus Dknk is a positive constant. In the case of smooth boundaries, the equation Dk = D − 1 holds, D being the dimension of the phase space. For fractal boundaries Dk can be larger, but obviously we always have Dk ≤ D. On the other hand, the number of boxes in the whole region of phase space, grows as Nk and N in αk = D − Dk is the uncertainty exponentBy following the method of the box-counting dimension ε; the term of the uncertainty exponent αk, is related with the fractality of the boundaries and contains the variation of the basin entropy with the box size; finally there is a term that depends on the number of different colors mk. All these terms depend on the dynamics of the system, while the scaling box size ε depends only on the geometry of the grid.This last expression reveals important information. The basin entropy has three components: the term αk = 1) and fractal boundaries (αk < 1). For both of them, smooth and fractal basins, we get Sb → 0 when ε → 0, but it converges faster in the smooth case. That is, it is more difficult for the basin entropy to decrease its value in a system with fractal boundaries. Despite other important factors, fractal boundaries introduce a larger uncertainty than the smooth ones. Furthermore, if αk = 0 then Sb > 0 no matter the scaling box size . In Section 4.1, we will show on an example that the criterion (8) enables reliably to find parameter regions exhibiting fractal boundaries. A detailed proof of the log2 criterion can be found in the This is a sufficient but not necessary condition: as we shall discuss in Section 4.1, there may be fractal boundaries with αk and the number of attractors NA.Here we illustrate the main features of basin entropy with several examples of dynamical systems, showing how its dependence on the boundary size The term To study the contribution of this term, we consider the damped Duffing oscillator given byδ the faster the orbits tend to the fixed points and, as a consequence, the basin of attraction appears more deformed for smaller values of δ  of the ues of δ . The damδ, it is noticeable that the basin of ε-box would be more difficult to predict in the second case. Nevertheless, both basins have the same uncertainty exponent α = 1 since in both cases the boundary is smooth. The differences in the values of the basin entropy originates from the differences in the region of discretized phase space occupied by the boundary, that is, the boundary size, which is reflected by the term Observing the basins of attraction corresponding to the three different values of Sb versus the scaling box size ε for three different values of the damping coefficient δ. The results are shown in the log-log plot of δ. We must note that we have normalized the region of the phase space, so that the values of the scaling box size ε in all the plots of the paper are the inverse of the number of pixels used as a grid. By taking logarithms on both sides of To highlight this effect, we have computed the basin entropy α = 1 and NA = 2 for all our simulations, it is clear that the variation of the basin entropy with δ is entirely due to the term α = 1 within the statistical error for all the fits. Therefore, although all these basins have the same uncertainty exponent, they have a different basin entropy for a given value of ε. The basin entropy is sensitive to their different structure and is able to quantify their associated unpredictability.Since in this case, we have αkThe fractal dimension of the boundaries also plays a crucial role in the formulation of the basin entropy. This is reflected in the uncertainty exponent E = 0.2 to E = 0.22, the fractal dimension of the boundaries is modified with E, though the Wada property is preservedwhich describes the motion of a particle in an axisymmetrical potential well that for energy values above a critical one, the trajectories may escape from the bounded region inside the well and go on to infinity through three different exits. If we vary the energy from E is a change of the slope in the log-log plot of α of the boundary. Smaller energies lead to smaller uncertainty exponents, since the boundaries have a more complex structure and consequently the slopes in the log-log plot decrease too. Obviously the offset also varies for the different values of the energy. This is related to changes in the boundary size α. This example shows that the scaling of the basin entropy with box size directly reflects the fractal dimension of the basin boundaries. For small box sizes this effect dominates and the largest fractal dimensions of the basins gives the largest basin entropies even though the offsets are different . The basins of attraction of this system have disconnected Wada boundaries, that is, all the basins share the same boundaries and are disconnectedwhere ε considered.From chaotic parameter setε, and then we evaluate the basin entropy associated to the corresponding basins of attraction for different parameter settings. We call the plot of the basin entropy in a two-dimensional parameter space basin entropy parameter set. To illustrate the possibilities of this technique, we study the periodically driven Duffing oscillator F and the frequency ω of the driving, and for each basin we compute its corresponding basin entropy. We have used a resolution of 200 × 200 boxes (ε = 0.005) with 25 trajectories per box (a million trajectories per basin) to compute the basins of attraction and the same region of the phase space Ω =  ×  for all the pairs .One of the most interesting applications of the basin entropy is to use it as a quantitative measure to compare different basins of attraction. We propose an analogy with the concept of F, ω) for different values of the forcing amplitude and frequency. The hot colors indicate higher values of the basin entropy, while the white pixels are for zero basin entropy. The set of parameters with zero basin entropy indicates that the basin of attraction has only one attractor. Although there is no uncertainty about the final attractor of any initial condition, trajectories may still be very complicated if the attractor is chaotic. This is actually the case for The result is presented in hottest point of the basin entropy parameter set corresponds to the basin of attraction shown in The i.e., 50000 trajectories for each value of , instead of the million trajectories needed for the usual procedure . Thus, we speed up the computation by a factor 20.Remarkably, it is also possible to explore the parameter space using only a few boxes instead of computing the high resolution basin for each parameter set. To infer a good approximation of the basin entropy, we applied a Monte Carlo sampling method. We have used 2000 boxes for each point in the parameter set, N, since the error decreases as To evaluate the discrepancies between the usual procedure and the random sampling we have calculated the relative error n 5% see . If highSbb in the parameter plane. This quantity reflects the uncertainty associated to the boundaries, and we have seen in Section 2 that if Sbb > log2 then the boundary is fractal. For a given scaling box size ε, this process cannot distinguish a true fractal boundary from a smooth boundary which at this scale separates more than two basins inside one box. The results for the periodically driven Duffing oscillator are depicted in the colormap of F, ω) displaying only one attractor. By means of this plot we can detect parameter regimes where boundaries are fractal, depicted with hot colors. sufficient but not necessary condition for fractal boundaries. Indeed, the log2 criterion works only for cases with three or more basins .


Parameter sets that represent infectivity of pathogens are estimated for EBOV infection and compared with influenza virus infection kinetics. The average infecting time of wild-type Vero cells by EBOV is slower than in influenza infection. Simulation results suggest that the slow infecting time of EBOV could be compensated by its efficient replication. This study reveals several identifiability problems and what kind of experiments are necessary to advance the quantification of EBOV infection. A first mathematical approach of EBOV dynamics and the estimation of standard parameters in viral infections kinetics is the key contribution of this work, paving the way for future modeling works on EBOV infection.The recent outbreaks of Ebola virus (EBOV) infections have underlined the impact of the virus as a major threat for human health. Due to the high biosafety classification of EBOV (level 4), basic research is very limited. Therefore, the development of new avenues of thinking to advance quantitative comprehension of the virus and its interaction with the host cells is urgently needed to tackle this lethal disease. Mathematical modeling of the EBOV dynamics can be instrumental to interpret Ebola infection kinetics on quantitative grounds. To the best of our knowledge, a mathematical modeling approach to unravel the interaction between EBOV and the host cells is still missing. In this paper, a mathematical model based on differential equations is used to represent the basic interactions between EBOV and wild-type Vero cells  Ebola was characterized for the first time in 1976 close to the Ebola River located in the Democratic Republic of the Congo WHO, . Since tFiloviridae, from Latin filum which means thread , the EBOV infection model is considered as follows:U) or an infected state (I). Cells are replenished with a constant rate λ and die with rate ρ. Note that the condition λ = U0ρ should be satisfied to guarantee homeostasis in the absence of viral infection, such that only ρ is a parameter to be determined. Virus (V) infects susceptible cells with rate constant β. Infected cells are cleared with rate δ. Once cells are productively infected, they release virus at rate p and virus particles are cleared with rate c.EBOV target cells can be either in a susceptible (U0) can be taken from the experiment in Halfmann et al.  is set to zero. The viral titer in Halfmann et al. (ffu/ml). The initial viral load (V0) is estimated from the data using the fractional polynomial model of second order  difference on log scale between the model output, ŷViral load variability is very large for several viral infectious diseases  and the data set in Halfmann et al. . The ideyi denotes the model variables, Θj is the parameter of interest, and the ratio wjΘ/wyi is the normalized factor corresponding to its nominal value . Moreover, EBOV infection has been adapted to guinea pigs and mice   has been broadly studied Xia, . All par has beenp and c show a convex shape of which the optimization routine can reach their minimum. Note that the profile of δ is flat in one tail, suggesting that parameter δ can be chosen arbitrarily small without affecting the fit quality  is slightly below other viral infection results  (Table p can be decreased given a higher effective infection rate (β), a replication rate of at least 31.8 ffu/ml cell−1 day−1 is still needed to achieve a good fit of the viral replication kinetics in Figure EBOV is known to replicate at an unusually high rate that overwhelms the protein synthesis of infected cells Sanchez, . Consist95% CI: 3−580  at 4.2 . The results of 1000 bootstrap replicates reveal that fixing the parameter c improves the fitting with a narrow confidence interval .Scatter plots are a graphical sensitivity analysis method, and a simple but useful tool to test the robustness of the results. The estimated parameters are plotted against each other. Scatter plots for the parameters in Figure in vitro and within a short period. Similarly, the effect of the infected cell death rate (δ) can also be neglected. This could be explained by the fact that the observed Ebola viral load was not decreasing  are sensitive, in the sense that a small change in parameter value can lead to a large difference in viral kinetics. Figure p, and c govern the infection kinetics while the effect of the two parameters ρ and δ can be neglected for this data set. Therefore, fixing both ρ and δ is adequate for the presented problem.The sensitivity study for the mathematical model Equations 1–3) is performed in a similar fashion to Brun et al. ; Soetaer is perfog Figure . The remp can be seen as consistently increasing the viral load because their respective sensitivity functions are always positive, in contrast to the parameter c. Note that the absolute magnitude of change in the sensitivity functions of these three parameters is approximately equal over time  while the susceptible cell population remains relatively constant, where r0 is the leading eigenvalue which solves the equation r20 + (c + δ)r0 − (βpU0 − cδ) = 0.Computational simulations for the best fitting of the proposed mathematical model Equations (1–3) plotted in Figure 7ffu/ml, which in general is 10 fold higher than those reported in influenza virus infection , which is defined as the expected number of secondary infections produced by an infected cell in its lifetime  and computed as follows  in Halfmann et al. . The iniEbola virus (EBOV) is highly pathogenic for humans, being nowadays one of the most lethal pathogens worldwide. Ebola fatalities are predominantly associated with uncontrolled viremia and lack of an effective immune response  . Known influenza virus titers were incubated without target cells and followed up to determine the remaining infectious titers  with an unspecific limitation of resources. However, a logistic model can explain only the growth behavior of the virus. As pointed out before . Future modeling studies need to quantify the situation in vivo where the effect of the immune system is taken into account.The in vitro. In addition, due to the critical relevance of the cytokine effects in EBOV pathogenesis, future modeling attempts should be directed to establish a more detailed model of interactions between the relevant cytokines and EBOV. Further insights into immunology and pathogenesis of EBOV will help to improve the outcome of this lethal disease.The modeling work developed in this paper paves the way for future mathematical models and experiments to shed light on the reasons for less efficient control of Ebola virus infections. Determining empirically the EBOV clearance rate in the absence of target cells would fulfill the picture of EBOV kinetics The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


For images containing indexable spots, the intensities of the expected reflections and their variances are obtained by profile fitting after mapping the contributing pixel contents to the Ewald sphere. The varying intensity decline owing to the angular distance of the reflection from the surface of the Ewald sphere is estimated using a Gaussian rocking curve. This decline is dubbed ‘Ewald offset correction’, which is well defined even for still images. Together with an image-scaling factor and other corrections, an explicit expression is defined that predicts each recorded intensity from its corresponding structure-factor amplitude. All diffraction parameters, scaling and correction factors are improved by post-refinement. The ambiguous case of a lower point group than the lattice symmetry is resolved by a method reminiscent of the technique of ‘selective breeding’. It selects the indexing alternative for each image that yields, on average, the highest correlation with intensities from all other images. Processing a test set of rotation images by XDS and treating the same images by nXDS as snapshots of crystals in random orientations yields data of comparable quality, clearly indicating an anomalous signal from Se atoms.A functional expression is introduced that relates scattered X-ray intensities from a still or a rotation snapshot to the corresponding structure-factor amplitudes. The new approach was implemented in the program  It is expected that this new X-ray source will strongly contribute to our knowledge of the large group of biological objects such as membrane proteins that are difficult to grow as macroscopic crystals.Destruction of the irradiated object by a single pulse is much slower than the pulse duration, so that data can be collected at room temperature before the sample deteriorates owing to radiation damage. As a consequence many diffraction snapshots are required, each from a different sample in a random orientation, to yield a complete data set. For crystalline samples diffraction data are partially recorded on still images, which presents a challenging task to processing that traditional crystallographic programs were not designed to handle.CrystFEL . These programs rely on the Monte Carlo method for estimating reflection intensities, thereby assuming that other factors such as fluctuations in the incident-beam intensity, wavelength and spectrum as well as the irradiated volume of the specimen ‘integrate out’ upon averaging  et al., 2012CrystFEL generates a perfectly twinned data set of higher symmetry by merging all snapshots, which renders the subsequent structure solution more difficult. Recently, new methods for breaking the indexing ambiguity have been described  pointing from the X-ray source towards the crystal.(i) The monochromatic incident-beam wavevector m2 passing through the origin of the laboratory system. During X-ray exposure the crystal is uniformly rotated by some angle Δϕ around this axis. Still images are treated as a limiting case where Δϕ = 0 and the concept of a rotation axis loses its meaning.(ii) The fixed rotation axis b1, b2, b3} of the single crystal and the associated reciprocal basis h, k, l are integers.(iii) The right-handed set of unit-cell basis vectors {detector system {D1, D2, D3} imagined to be fixed in the instrument and translated by the origin vector D0.(iv) The right-handed orthonormal The detector consists of one or several rectangular, planar X-ray-sensitive segments. Their position and orientation is specified with respect to the detector system, which renders the specification independent of any detector movements. For a well calibrated instrument this greatly reduces the number of parameters that need to be refined during data processing.Any right-handed orthonormal segment system {d′1, d′2, d′3} and an origin vector d′0 are specified for each segment with respect to the detector coordinate system. The X-ray-sensitive area consists of identical pixels of sizes QX, QY (mm) along d′1 and d′2, respectively.A right-handed orthonormal d′0 in terms of {d′1, d′2, d′3}, the location x′ of a segment pixel x, y with respect to the detector system is given by laboratory coordinates x of the same pixel x, y are F| from the crystal. F has a negative sign if d3 points towards the crystal.Expressing the segment origin M, and the combined effects of beam divergence and mosaicity, σD.For accurate integration, the spot shape and extent are modelled as Gaussian distributions involving two parameters: the standard deviations of the reflecting range (mosaicity), σ3.nXDS was developed for automatic determination of scaled and fully corrected reflection intensities from monochromatic X-ray diffraction images, each from a different crystal in a random orientation. nXDS uses many ideas, routines and the overall structure of the rotation data-processing program XDS . To assure that t is non-collinear with S0, one of its components is reset to 0, namely the component corresponding to the maximum absolute value of the components of S0. If this is not unique, the first occurrence of the maximum is taken. For example, t =  if the y coordinate is the first occurrence of the absolutely largest component of S0. A right-handed orthonormal system {t1, t2, t3} can then be defined by x, y in the COLSPOT step a scattering vector S can be calculated, D0.A plane with the incident-beam vector 3.5.For each snapshot this step tries to find a crystal lattice that explains the observed diffraction spots by assigning indices to them and refines all parameters. The step returns a list of the successfully processed images and the indexed spots as well as the corresponding diffraction parameters. These parameters are refined for accurate prediction and integration of all reflections that are expected to occur in each snapshot.et al.  pointing from the X-ray source towards the crystal. Diffraction occurs along the wavevector S when the crystal is rotated so that p* on the Ewald sphere satisfying the Laue equationsWe would like to find the smallest rotation that moves a reciprocal-lattice point onto the Ewald sphere. Let bS0, S0| and Extending earlier work  must vanish and the matrix of second derivatives must be positive definite, f, the Laue equations are satisfied if 1, Δ2, while the negative sign corresponds to the maximum distance from the Ewald sphere. Using the abbreviations p* on the Ewald sphere reachable by a rotation of Δ, S, p* all lie in the plane defined by p* by the shortest path if it is rotated about an axis parallel to the plane normal and passing through the origin of reciprocal space. The rotation axis is identical to the ‘β-axis’ .In the case of the rotation method the crystal is forced to rotate about a fixed axis Finally, the angular deviation of the reciprocal-lattice point 3.5.2.j enumerate the n strong spots with observed centroids jX′, jY′, detector segment identifying number js and reflection indices jh, jk, jl. To each j, we denote the reciprocal-lattice point Sj, an angular deviation from the Ewald sphere τj and reflecting-range expansion factors ζj can be determined as described in the previous section. With the detector segment recording the strong spot at distance laboratory system)For each image, let E becomes a quadratic function of these changes. Minimization then leads to a system of normal equations whose solution is used to update the parameters. During refinement, the cell metric obeys constraints imposed by the lattice symmetry by choosing an appropriate set of free parameters for representing the allowed changes of the basis vectors (details not shown).The residuals are expanded to first order in the parameter changes so that wX, wY, wτ until convergence is reached. New weights are then determined so that wX, wY, wτ.The minimization procedure is repeated keeping the same set of weights 3.6.CORRECT step. No correction factors are applied except for compensating missing parts in the reflection profile owing to overlap with bad pixels or closely neighbouring reflections.Starting with the refined diffraction parameters for the successfully indexed data images, this step determines the recorded reflection intensities by two-dimensional profile fitting and saves the results on file for subsequent processing by the XDS  describes the expected intensity profile of the reflections mapped to their specific ∊1, ∊2 plane tangential to the Ewald sphere. Initially, Gaussians are assumed with the parameter σD modelling the combined effects of beam divergence and mosaicity as D is estimated from the observed variance in intensity of scattered rays for the strong reflections and provides information on the spot width. In a second step the initial Gaussian form for ω12 is replaced by the superposition of the profiles of all strong reflections and is used for definition of the final integration region included in profile fitting of all reflections predicted to occur in the diffraction image.Because of crystal mosaicity and beam divergence, the intensity of a reflection is smeared around the diffraction maximum. Following earlier work , models the dependency of the reflection intensity on the angular distance from the surface of the Ewald sphere as a Gaussian with the mosaicity σM of the crystal as the standard deviation. An estimate of σM is found as described previously  reflections as obtained from the previous step are scaled and corrected by a modified post-refinement procedure and then saved in the final reflection output file for subsequent structure-solution software packages.As described below, the new concept of the Ewald offset correction factor as a replacement for partiality links raw intensities to structure-factor amplitudes, rendering classical post-refinement applicable even to still snapshots.3.7.1.INTEGRATE step, is proportional to the ‘true’ intensity I (the squared structure-factor amplitude),C consists of a factor T that accounts for different scale and resolution fall-off between the snapshots.A reflection intensity Q), Lorentz factor (L), polarization (P), air absorption (A) and sensor-thickness correction (O). These corrections are functions only of the diffraction parameters of the snapshot that recorded the reflection.Ewald offset correction. The recorded intensities of the reflections of the same image would be directly comparable if they simultaneously obeyed the Laue equations, i.e. τ = 0 for all reflections. The idea is to correct the intensity by the estimated decline owing to the angular distance of the reflection from the surface of the Ewald sphere. Using the dimensionless variables t = τ/21/2σM and z = ∊/21/2σM (∊ = ζ·Δϕ/2) this leads to the definition of the Ewald offset correction z = 0). Q can be calculated for each reflection from the incident-beam wavevector So, the reciprocal basis M and the oscillation range Δϕ.Lorentz factor. Assuming an infinitely thin Ewald sphere, the Laue equations can be satisfied for a reciprocal-lattice point by a variation in the wavelength or the direction of the incident beam relative to the crystal. For an ideal crystal the scattered intensity is sharply concentrated around the direction of the diffraction maximum with a solid angle much smaller than the finite aperture of a detector pixel. Consequently, only integrated intensities can be observed: these are related to their squared structure factors by the Lorentz correction. Explicit forms of the corrections are available  Rotation method. For a fixed wavelength but a variable direction of incidence , the correction factor is(ii) Laue method. For a variable wavelength but a fixed direction of the incident beam, the correction factor is(iii) Powder method. For a fixed wavelength but a variation of the incident beam with two degrees of freedom , the correction factor isLorentz correction is always a simple function of resolution and therefore does not affect the agreement among intensities of symmetry-related reflections.Polarization. The intensity of scattering from the crystal is proportional to the polarization factor E0 denotes the electrical field vector of the incident beam and S the diffracted beam wavevector. If n denotes the polarization plane normal and p the probability of finding the field vector E0 in this plane, E0 is found with equal probability pointing along S0 × S or (S0 × S) × S0, so that et al., 1982n and p are provided by the user (not refined).Air absorption. The recorded intensity is reduced from its ‘true’ value owing to air absorption of the diffracted beam by the factor D is the distance (in millimetres) between the crystal and the position of the reflection spot on the detector segment. μ is a wavelength-dependent input constant (not refined).Sensor-thickness correction. The recorded intensity is increased owing to the effect of the oblique incidence of the diffracted beam on a sensor of finite thickness. Let δ denote the thickness of the sensor and κ the fraction of intensity loss per millimetre. If the diffracted beam makes an angle ω with the segment normal, the probability of a photon penetrating the sensor undetected is exp(−κδ/cosω). Thus, the correction Scaling and temperature factor. Using the parameters g and B, the factor g and B adjust differences in beam intensity and in crystal disorder and volume in a resolution-dependent way. This correction factor is obtained by comparison of symmetry-equivalent reflections from different images and therefore depends on a consistent indexing choice. For resolving possible indexing ambiguities, correlations between equivalent reflection pairs from different snapshots must be computed  For the given space group and unit-cell parameters all possible unique reflections within a given resolution range are generated. A prime number is determined that is slightly larger than twice the number of possible unique reflections and is used to allocate space for a hash table of this size.(ii) For each possible unique reflection a positive 64-bit integer is constructed from its unique indices and assigned a definite address in the table using the technique of hash-key transformations with quadratic probing to resolve key collisions  For each reflection from the snapshots the unique indices are determined and coded by their hash-table address, which is saved as an auxiliary reflection attribute. Thus, two reflections are symmetry-related only if they have identical hash-table addresses.(iv) Four auxiliary arrays of the size of the hash table are allocated: two for each data set to be compared. They are needed for calculating unique intensities and their variances for the reflections of the two data sets.Correlation. (i) Unique intensities and variances are estimated from symmetry-related reflections of the first snapshot by updating the contents of their associated hash addresses in the first two auxiliary arrays.(ii) For the intensity data from the second snapshot the procedure is repeated, this time updating the second two auxiliary arrays only if there is a positive entry from the first snapshot at the same hash address in the first two auxiliary arrays.(iii) Pairs of corresponding unique reflection intensities are obtained easily by scanning the second two auxiliary arrays for a positive contents.Thus, the total number of operations for calculating the correlation factor between one snapshot and all others is only proportional to the total number of recorded reflections of the snapshots.3.7.3.aFor a given space group and unit-cell parameters, a reciprocal cell basis in some reference orientation  For each snapshot all of its possible indexing choices are tested in succession, with the reindexed reflection intensities treated as a hypothetical reference data set. The mean value of the correlation factors with all other snapshots is calculated, and the running number of the reindexing choice yielding the largest correlation factor is saved. If several choices result in the same value for the correlation maximum, the first one in the list is selected.(ii) At the end of the cycle the list of optimal indexing choices just determined is used and replaces the previous selection. For some snapshots the new running number of the reindexing choice may differ from that of the previous cycle. If none of the snapshots needs to be assigned to a different indexing choice than before, the cyclic procedure terminates.This procedure usually terminates within ten cycles. The moderate amount of storage needed is proportional to the number of snapshots and to the size of the hash table. In addition one array is needed that keeps the hash-table address of the unique indices for each reflection of the whole set of images. This speeds up the computation of correlation factors between all pairs of snapshots. Within each cycle these computations can be carried out very efficiently and in parallel by a team of processors.3.7.5.We assume that all reflections have been consistently indexed. The reflection intensities of each image are still on different scales owing to differences in the intensity of the incident beam or irradiated crystal volume. Determination of the scale factors by the method described here is based on intensity estimates for the unique reflections occurring on each image, h enumerate the hn different unique indices of the reflections involved in scaling and l enumerate the ln images from which the reflections come. Let j enumerate the n unique reflections included. To each j, we denote intensity jh and image jl. The goal of the scaling procedure is to find factors lg > 0 and mean intensities hI > 0 by minimizing the target function . In each following cycle new scaling factors are found as c chosen to maximize the reduction of the target function. Expansion of the quadratic target function at the point G to G′ changes the mean logarithmic intensities by G′, J(G′) yields a > b, which leads to an optimal step size c = a/(a − b), so that the target function is reduced by Ψ − Ψ = ac.Let For the gradient, we have lG and subtracts an appropriate constant vector from hJ. This amounts to just changing the common scale of all images in the original problem, which is of no importance as we are only interested in the relative scale.Obviously, the solution thus obtained is not unique because the target function does not change its value if one adds an arbitrary constant to the logarithmic scaling factor It may happen that several sets of images exist that are not connected by common measurements. In this case the target function could be thought of consisting of a sum of the same type of target functions, one for each unconnected subset of images. Now there is an arbitrary common factor for each subset. Apparently, the presence of arbitrary common factors for each subset of images does not prevent convergence of the cyclic solution procedure described here.3.7.6.IDXREF and INTEGRATE steps to minimize deviations between the observed and the predicted locations of the strong spots and to minimize their angular distance from the Ewald sphere. The angular part of the target function takes care of the fact that reflections can be visible only if they are close to the Ewald sphere. In addition, the distribution of τ angles thus obtained provides an initial guess for the crystal mosaicity σM. However, the angular part of the initial refinement target cannot account for the fact that very strong reflections can still be observed even if they are farther away from the Ewald sphere than the weaker reflections. This leads to a systematic bias in the initial parameter refinement so that strong reflections will be predicted to be closer to the Ewald sphere than they really are.As mentioned above, the diffraction parameters of each image are refined in the et al., 1979et al., 1985Q defined above that is applicable for rotation images as well as stills. The ‘post-refinement’ variant implemented here considers the possible unique reflection intensities as free parameters that are to be refined along with the diffraction parameters of each snapshot  detector parameters.Global parameters.hI, the squared structure-factor amplitudes for the possible unique reflections h .(ii) Global constants.n and p, the polarization plane normal and the degree of polarization.μ, the fraction of intensity loss per millimetre in air.δ, the thickness of the detector sensor.κ, the fraction of intensity loss per millimetre in the sensor.ϕ, the rotation axis and oscillation range (Δϕ = 0 for ‘stills’).(iii) The target function hI, keeping the current parameter values constant. Minimization of the target function yields Xw, Yw, Iw.Each refinement round starts with the determination of the unique reflection intensities E becomes a quadratic function of these changes. Minimization then leads to a system of normal equations whose solution is used for updating the parameters. The gradients of ΔXj, ΔYj and jC are computed from analytic expressions (not shown).During a refinement round the diffraction parameters for each snapshot are corrected iteratively to minimize the target function until convergence is reached. The residuals are expanded to first order in the parameter changes so that Fortunately, these calculations are independent for each snapshot and can be performed in parallel by a team of processors. Moreover, the memory requirements are almost negligible even when the refinement of detector parameters is included.4.nXDS in comparison to conventional data reduction by XDS, 20 000 consecutive rotation images were collected at 100 K from a crystal of a selenomethionine-labelled double mutant of the RNA-processing factor SCAF8  located at 300 mm distance. The crystal has P43 space-group symmetry, which is lower than the 422 lattice symmetry, implying a twofold indexing ambiguity.As an example to demonstrate the quality of data processed by XDS as conventional rotation data. The lower part shows the corresponding quantities as obtained from nXDS. Here, reflections were only included if their Ewald offset correction was larger than 0.7. A total of 356 854 reflections in the resolution range 15–2 Å were integrated by XDS, so that for each unique reflection almost eight symmetry-related reflections are available. Because contributions to each reflection are also recorded by adjacent images, it is not surprising that nXDS found almost ten times more reflections, which is consistent with the correspondingly higher multiplicity of observations.The processing results are summarized in Table 1I/σ(I)〉 would come out about the same regardless of whether the images were processed by XDS or nXDS. This is not the case: the mean signal-to-noise ratio is higher by a factor of almost three for the results from XDS. The lower accuracy of nXDS presumably results from two-dimensional instead of three-dimensional profile fitting and the lack of other corrections not carried out yet by this version of nXDS. The nearly perfect correlation factors CC1/2  and heavy-atom peak heights.Phasing of single anomalous diffraction data was performed for the e Table 4 and the 5.nXDS that has borrowed many ideas and routines from the rotation data-processing package XDS as well as from the powerful post-refinement technique that has been in widespread use for several decades.This study describes a new approach for processing a large set of snapshots from randomly oriented crystals that does not rely on the Monte Carlo method of integration. Instead, the concept of the Ewald offset correction factor was devised to overcome difficulties arising from the use of partiality for modelling reflection intensities recorded by snapshots. The new approach has been implemented in the program The implemented Ewald offset correction relies on a Gaussian model for the rocking curve, assuming a sufficiently large crystal whose shape transform can be ignored. In this case the exact functional form of the curve is not critical for reflections sufficiently close to the Ewald sphere. In the test data set, a reflection was only included if its Ewald correction factor was larger than 0.7.nXDS delivers results almost approaching those obtained by XDS and is able to retrieve the anomalous signal from a selenomethionine-labelled protein crystal. The source of the lower accuracy of nXDS is not yet clear. It could result from two-dimensional instead of three-dimensional profile fitting and the omission of information from weak contributions to reflections further away from the Ewald sphere that are used only by XDS. In fact, a small improvement in overall data quality by 0.9% in 〈I/σ(I)〉 and 2.9% in CCano was observed upon the inclusion of weaker contributions when the minimum required Ewald offset correction was lowered from 0.8 to 0.7.As shown for the test case with fine-sliced rotation images of excellent quality, nXDS. These data typically vary for each snapshot in wavelength, bandwidth and crystal parameters. Although nXDS allows some of these to change for each snapshot, program modifications are likely to become necessary when the incident-beam bandwidth can no longer be substituted by a mean wavelength or if the shape transform of the crystals cannot be ignored. Presently, nXDS can only accept images that XDS can read. Work is in progress to adapt the package to also handle the detectors used at FEL beamlines and to make nXDS and its documentation available from the internet.So far, no ‘real’ FEL data have been processed by