When learning to perform a novel sensorimotor task, humans integrate multi-modal sensory feedback such as vision and proprioception in order to make the appropriate adjustments to successfully complete the task. Sensory feedback is used both during movement to control and correct the current movement, and to update the feed-forward motor command for subsequent movements. Previous work has shown that adaptation to stable dynamics is possible without visual feedback. However, it is not clear to what degree visual information during movement contributes to this learning or whether it is essential to the development of an internal model or impedance controller.We examined the effects of the removal of visual feedback during movement on the learning of both stable and unstable dynamics in comparison with the case when both vision and proprioception are available. Subjects were able to learn to make smooth movements in both types of novel dynamics after learning with or without visual feedback. By examining the endpoint stiffness and force after learning it could be shown that subjects adapted to both types of dynamics in the same way whether they were provided with visual feedback of their trajectory or not. The main effects of visual feedback were to increase the success rate of movements, slightly straighten the path, and significantly reduce variability near the end of the movement.These findings suggest that visual feedback of the hand during movement is not necessary for the adaptation to either stable or unstable novel dynamics. Instead vision appears to be used to fine-tune corrections of hand trajectory at the end of reaching movements. In order to perform motor tasks, the central nervous system integrates multiple modes of sensory information It has been shown that people are able to learn the novel dynamics of both stable environments as well as unstable or unpredictable environments While both visual and somatosensory feedback are likely to play a critical role in the learning of novel dynamics, several previous studies on motor learning have shown that adaptation to stable dynamics is possible even when subjects are provided only with delayed visual feedback of the trajectory Eight  neurologically normal subjects participated in the study (mean age: 25±4 years). All subjects were right-handed according to the Edinburgh handedness inventory Subjects were seated with their shoulders restrained against the back of a chair by a shoulder harness. A custom-molded rigid thermoplastic cuff was securely fastened around the subjects' right wrist and forearm, immobilizing the wrist joint. Only the shoulder and elbow joints remained free to move in the horizontal plane. The subjects' forearm was secured to a support beam in the horizontal plane and the cuff and beam were coupled to the handle of the parallel-link direct drive air-magnet floating manipulandum (PFM) used to generate the environmental dynamics . Movemeny-axis corresponding to the straight line from the shoulder joint to the target circle and with the positive x-axis corresponding to the line from the shoulder joint to the right direction. The subjects' view of the PFM and arm was blocked by an opaque tabletop positioned above the arm and the PFM. The start and target circles were projected on to the tabletop by an overhead projector throughout all of the experiments. The projector was also used to display a 0.5 cm diameter circular cursor used to track instantaneous hand position when appropriate for the visual conditions as described below. A computer monitor positioned beyond the PFM in front of the subject provided knowledge of results about movement duration  and final hand position . If a subject received “OK” for all parameters then the trial was considered successful. Movement duration was considered OK within the range 600±100 ms. The movement duration was constrained to be within this range for all parts of the experiment. Subjects' hand position was displayed with the cursor before the start of each trial in order to facilitate movement to the start circle. Once the cursor was moved within the start circle a trial was initiated by three beeps spaced at 500 ms intervals. Subjects were instructed to begin movement on the third beep and to try to reach the target circle by the fourth beep, 600 ms later. Two additional beeps spaced 500 ms apart were heard once the target circle was reached to indicate how long subjects had to hold a steady hand position within the target circle. No instructions were given to the subjects about the trajectory that they should perform in order to complete the task. Subjects were only instructed that they were required to perform successful movements in order to complete the experiment.Movements investigated in this study were right-handed forward reaching movements in the horizontal plane at subjects' shoulder level. Movements were made from a 2.5 cm diameter start circle centered 31 cm in front of the shoulder joint to a 3 cm diameter target circle centered 56 cm in front of the shoulder joint. The origin of the coordinate system was centered at the shoulder with the positive xF, yF) depended on the position of the subjects' hand (x) relative to the y-axis. β was chosen as 400 N/m or 300 N/m for male and female subjects respectively so that the forces experienced would be relatively strong. These values were chosen based on previous work using this type of force field ẋ, ẏ). This stable force field would produce a stable interaction with the subjects' limb. All subjects experienced each environmental condition under both visual feedback and no visual feedback conditions.Learning and stiffness were examined under three different environmental dynamics or force fields. These three environments were: a null force field (NF) (baseline), an unstable position-dependent divergent force field (DF)  where anThis experiment was performed to test the effect of visual feedback on learning and final stiffness after adaptation to presentation of novel environmental dynamics. Two feedback conditions, online visual feedback and no online visual feedback, were presented. For online visual feedback, the cursor representing hand position was shown throughout the entire trial. This cursor was a 0.5 cm diameter circular cursor projected directly over top of the subject's hand, tracking the instantaneous hand position. This was updated at a refresh rate of 60 Hz. For the condition of no visual feedback, as soon as the movement was initiated the tracking cursor was turned off for the duration of the movement. Once the final two beeps were heard (1000 ms after the end of the movement) the cursor was displayed again so that subjects could see their final hand position and move the hand back to the start circle to begin the next trial. No information about the actual trajectory was presented to the subjects. The start and target circles were displayed throughout the entire experiment under both visual conditions.Subjects were randomly divided into two groups (four subjects in each group): vision and no-vision. The no-vision subjects were exposed to each environmental condition under the no visual feedback condition first before experiencing each environmental condition under the visual feedback condition. The vision group was exposed to each environment under the visual feedback condition first and then under the no-vision condition. In both groups the NF field was experienced first before the other two force fields so that subjects could get accustomed to the PFM. The order of presentation of the DF and CF fields was balanced between subjects in each group. This design of the experiment has subjects adapt to the same force fields both with visual feedback and without feedback. One confounding factor in this design is that there could be some retention of what was learned in the first condition which could affect performance on the second condition. While this effect may be partially limited through the random presentation of force fields it could still affect factors such as speed of learning. However, by having subjects adapt under both conditions, direct comparisons of endpoint forces, trajectory variability, and particularly endpoint stiffness, which have large components of individual variability, can be directly compared across the conditions using repeated measures. Subjects performed both the vision and no vision experiments so that the stiffness measurements can be accurately compared across the conditions. This would not be possible using a design where different groups of subjects performed learning under each condition.Prior to the beginning of the experiment, all subjects trained for one day in the NF in order to adapt to the natural dynamics of the PFM itself as well as accustom them to the task constraints. On these pre-experimental days, full visual feedback of the subjects' trajectory was provided. The experiment was conducted over a period of six days for each subject so that only one of the three force fields under one of the two feedback conditions was ever experienced on a particular day. Each daily experimental session consisted of two parts: learning and stiffness estimation. In the learning phase subjects first performed 30 successful movements in the NF, where a successful movement was considered as one which ended within the allotted time and within the target circle. After 30 successful trials, the force field was activated. Subjects were unaware as to the start of the force field activation as no information was given about the number of trials before the force field would be activated. Subjects were then required to practice in the force field until 100 successful movements were performed. All trials were recorded whether successful or not. The stiffness measurement phase followed the learning phase after a short 5 minute break (see stiffness estimation below). Note that for the NF only the stiffness estimation phase was performed since the learning of novel dynamics was not required in the NF. All subjects had previously trained with the manipulandum in the NF field.We measured stiffness in the CF and DF force fields after extensive learning, as well as in NF movements. Full details of the stiffness estimation procedure can be found elsewhere y-axis). This measure of learning was used since movements in the NF as well as movements after adaptation in the force fields tended to be relatively straight. The absolute hand path error wasx (t) and ẏ (t) represent the shoulder-joint centered Cartesian x-position and y-velocity of subjects' hand respectively, while ot was taken 300 ms prior to the y-velocity crossing a threshold of 0.05 m/s and ft represents the time of movement termination (time when curvature exceeded 0.07 mm−1). To take into account the effect of switching off the DF once subjects crossed the 5 cm safety boundary, movements were assumed to remain at the boundary for the duration of the movement from the time it was crossed for calculation of the handpath error. To test whether learning occurred, a separate ANOVA was performed for each field and visual condition to compare the absolute hand path error on the first and last 10 trials of learning. Each ANOVA had a main effect of Learning , and random effect Subjects.Learning in the different force fields was calculated as the error relative to the straight line joining the start and end targets , an ANOVA with main effects of Vision , Field (type of force field), and Order (order of learning), and random effect Subjects was performed on the data. Second level interaction terms were also examined for (Field*Vision) and (Field*Order). Separate ANOVAs for the CF learning rates and for the DF learning rates were performed in order to further examine the Order effect in the two fields. The ANOVA had main effects of Vision  and Order (order of learning), and random effect of Subjects.In order to compare learning, exponential curves were then least-square fitted to the hand path errors for each subject in order to compare learning rates as a function of trial number (using Matlab® R14 function lsqcurvefit). The exponential curve took the form ofThe speed of learning across the force fields and visual conditions was also examined by looking at the number of trials required in order to complete 100 successful trials. A separate non parametric Kruskal-Wallis test was performed on the results of each force field with grouping variable Vision .l = a/b, where a is the maximum orthogonal deviation of the trajectory from the straight line joining the start and end targets and b is the distance of this line. The linearity ratio was determined using the mean trajectory of the last 20 successful trials of learning averaged across all subjects for each field and visual condition. The linearity ratio was calculated in order to examine differences in trajectories between the vision and no vision conditions in each field as in The linearity ratio x-direction as a function of y-position was also calculated. This measure, independent of the actual trajectory chosen, was estimated across all subjects for each field and visual condition for the last 20 learning trials whether successful or not. However, to prevent single trials from unduly influencing the results, selected outlying trials were removed from the data if their path was far outside the trajectories of the rest of the trials. The criterion for removal was if the trial trajectory went outside 2.5 * the standard deviation of the twenty trials. If trials were removed then the next trial was included such that a total of 20 trials were used for each subject in each force field. In the NF field, 4 trials were removed out of the 320 trials total . In the CF, 25 trials out of the 320 trials were removed . In the DF, any trials which exited the safety zone were removed automatically. Out of the trials remaining, 6 trials out of the 320 trials were removed . The y-position data was transformed to a percentage of the maximum y-position of each individual trial so that all movements were aligned on the same scale since each movement terminated at a slightly different position. The corresponding x-position value in Cartesian space was found at each 1% increment of the y-position with the y-position ranging from 0–100% of the total movement. Variability was then calculated in the form of standard deviation for each individual subject and then averaged across all subjects. The movement variability was examined in order to determine the effects of vision. Significant differences between the standard deviation in the visual and non-visual feedback conditions were examined using paired t-tests with Bonferroni correction for multiple comparisons. At each 1% of movement distance from 2% to 98% of the movement distance, five points (±2%) were included in the paired t-test. Significant differences were considered at the 0.05 and 0.001 levels.When making repetitive movements there is always variability in the individual trajectories. In order to examine differences in the variability of these movements across the conditions, the mean standard deviation of movement trajectories in the Endpoint forces (measured at the handle of the manipulandum) from the last 20 successful trials during the learning phase of the experiment for the CF and DF were analyzed. The last 10 successful trials were used in order to look at endpoint forces after adaptation to the novel dynamics. Since there was no learning phase for the NF, the last 10 successful pre-trials during the stiffness estimation phase of the experiment for the NF were used instead. Mean endpoint forces for each subject at the mid-point of movement  were examined to determine if subjects adapted to the force fields with or without vision using different endpoint forces. The time interval of stiffness estimation was chosen for this comparison so that it would be possible to examine whether any differences in stiffness between the visual conditions could have be explained by changes in endpoint forces. An ANOVA with main effect of conditions , and random effects of subjects was used to examine this. If a significant main effect of conditions was found then a post-hoc test (Tukey's HSD) was used to test for significant differences across the conditions.K was estimated for each subject and condition. Endpoint stiffness was estimated by performing a linear regression on the mean change in endpoint force  and the mean change in position  during the 50 ms interval in the last half of the hold phase of the perturbation window. The relationship is shown in the equation: xxK, xyK, yxK, yyK) an ANOVA was performed with main effect condition  and random effect Subjects. If a main effect of condition was found to be significant, differences in stiffness between the visual feedback and no visual feedback conditions were examined using post-hoc tests (LSD).Using the force and position data recorded during the stiffness estimation phase of the experiment, the 2×2 endpoint stiffness matrix x- or y-axes after adaptation to the DF, the ratio of the stiffness increase in the xxK and yyK terms between the DF and NF fields was calculated. The relative increase in the xxK term was compared to the relative increase in the yyK term and tested with a paired t-test. This was performed separately for both the visual feedback condition and the no-visual feedback condition. For the NF condition, the mean of the visual and non-visual conditions was used.In order to examine whether stiffness was primarily increased in the Statistical analysis was performed using SPSS 10.0  with the exception of the multiple paired t-tests examining the variability of movements which were performed in Matlab® R14. Statistical significance was considered at the 0.05 level for all statistical tests. ANOVAs were examined in SPSS using the general linear model. When post-hoc tests were performed after a significant main effect was found, Tukey HSD test was used in all cases except for the testing of stiffness. For the testing of stiffness, the more liberal Fisher LSD post-hoc test was used. LSD was chosen, rather than a more conservative test such as Tukey's HSD, because it is more robust to a Type II error. In the case of stiffness results it was important to avoid cases where no difference between the two visual conditions was reported when there was a real difference between the conditions. The data was examined to determine if it came from a normal distribution using either the Kolmogorov-Smirnov test with Lilliefors significance correction or the Shapiro-Wilk test, depending on the degrees of freedom. The data was also visually inspected to confirm that it did not appear to depart from a normal distribution with homogeneous variances. Most data sets appeared to be from a normal distribution allowing for the use of parametric statistics. However, both the linearity ratios data set and numbers of trials required to achieve 100 successful trials data set were tested with the non-parametric Kruskal-Wallis test.Subjects initially made movements in the null field condition either with or without vision. Under these conditions, all subjects were able to make roughly straight movements of appropriate speed to the target. When the force fields were unexpectedly applied, subjects' trajectories were disturbed by the change in forces experienced. Gradually subjects were able to reduce the disturbance created by the novel dynamics by learning to compensate for it. F = 4.987; p = 0.027)  = 15.577; p<0.001). Similarly in the CF the last ten trials had lower handpath errors both with vision  = 50.57; p<0.001) and without vision  = 104.28; p<0.001). Subjects were able to adapt to or learn both stable and unstable dynamics regardless of whether visual feedback was given or not.To further confirm that learning occurred under both visual conditions in the force fields we examined the handpath error. The handpath error is a measure of the area between the actual trajectory and the straight-line path joining the start and end targets. A clear learning effect across all subjects is seen by examining the mean absolute hand-path error for the visual feedback and no visual feedback groups in the DF and the CF . At the F = 18.719, p<0.001) and Order  = 9.368, p = 0.006) but not for Field  = 1.855, p = 0.189). A significant interaction between Field and Order was also obtained  = 14.709, p = 0.001), whereas the interaction between Field and Vision was not significant  = 0.109, p = 0.745). The learning rates are shown in F = 0.548, p = 0.487) whereas in the CF there was a significant Order effect  = 23.479, p = 0.003). These results indicate that subjects were able to reduce the handpath error quicker the second time they adapted to the CF field whether this was with or without visual feedback. This was not true of the adaptation to the DF where there was no significant change in the learning rate the second time subjects adapted to the field.As learning occurred under all conditions, we next examined the speed of learning. In order to compare learning speeds under the various conditions, both the learning rate and the total number of trials required for success were examined. The learning rates were obtained by least squares fitting of the exponential curves to the handpath error data for each subject. The learning rates across all conditions were tested with an ANOVA. Significant main effects were found for Vision (Chi-Square(1) = 3.982; p = 0.046) and CF  = 7.467; p = 0.006) force fields.Different numbers of trials were required in order to complete 100 successful trials under the various conditions. The mean trials across subjects (±standard deviation) were: DF vision = 134.3±34.8; DF no vision = 191.4±82.5; CF vision = 107.0±3.5; CF no vision = 159.8±31.5. There were significantly fewer trials required with visual feedback in both the DF  = 7.329; p = 0.007) condition  = 12.687; p<0.001). These results are consistent with previous findings Chi-Square(1) = 0.013; p = 0.911).Although adaptation occurred under both the vision and no-vision conditions, upon closer examination of movement trajectories, some differences in the kinematic features were uncovered. The mean trajectory used to reach the target location was examined with the linearity ratio. The ratio was found to be slightly larger for the no vision condition in both the NF and the CF, indicating that the movements were more curved under the no-visual condition in these two force fields. In the NF, linearity averaged across all subjects was 0.0285±0.0126 and 0.0332±0.0154 for the visual and no visual conditions respectively. In the CF linearity was 0.0405±0.0197 with vision and 0.0488±0.0228 without vision. In both force fields, the movements in the visual condition were significantly more linear than those in the non-visual condition  = 157.0, p<0.001). Using Tukey's HSD post-hoc test, it was determined that there were two distinct groups: the two CF conditions (vision and no-vision) were significantly different than the other four conditions (DF and NF) . The two CF conditions were not significantly different (p = 0.71). Similarly there were no significant differences across the visual conditions in either the DF (p = 0.97) or NF fields (p = 0.75). None of the other possible comparisons were significantly different either . There was also no significant main effect for the conditions in the y-axis  = 1.25, p = 0.31).After adaptation to the force fields, the endpoint force that the subjects produced against the robotic interface was examined . SimilarAfter learning was finished in each condition, subjects' endpoint stiffness was estimated using controlled position perturbations F = 2.907, p = 0.027), shape  = 4.274, p = 0.004), and size  = 11.054, p<0.001), post-hoc tests (LSD) were used to examine differences between the visual conditions. While differences occurred between the three force fields, this modification of stiffness has been extensively examined in previous work p = 0.005). Similarly, there was a significant difference in the shape between the visual and non-visual conditions (p = 0.048) but not in the size (p = 0.525). In the DF, the endpoint stiffness was significantly increased in size (p = 0.031) for the no vision condition compared to the visual condition. However, there were no significant differences in either the orientation (p = 0.969) or shape (p = 0.891) between the visual and non-visual endpoint stiffness. Finally in the CF, there were no differences in the size (p = 0.898), shape (p = 0.526) or orientation (p = 0.506) of the endpoint stiffness ellipse between the visual and non-visual conditions.To determine the overall effect of vision on the endpoint stiffness of the limb the mean endpoint stiffness ellipses averaged across all subjects for the three different dynamics and under the vision and no vision conditions were examined . DifferexxK term  = 29.909, p<0.001) and yyK term  = 4.697, p<0.001) were found. There were no significant main effects found for the xyK  = 1.555, p = 0.198) and yxK  = 0.599, p = 0.701) terms of the stiffness matrix. The differences in the xxK and yyK terms were investigated further with post-hoc tests. There were no significant differences between the visual conditions in the NF force field (p = 0.903) or the CF force field (p = 0.998). However the xxK term was significantly larger in the no-vision condition compared to the vision condition in the DF (p = 0.033). There was also no significant difference in the yyK term between the visual conditions in the CF (p = 0.530).There were however significant differences in the yyK term for both the NF and DF force fields. In the NF, the yyK in the visual condition was larger than that in the no-vision condition (p = 0.021) whereas in the DF, the yyK term in the no-vision condition was larger than that in the visual condition (p = 0.036).The four elements of the stiffness matrix were also examined for confirmation of the previous results. A significant main effect for both the xxK-component of stiffness, however a significant difference was found in the yyK-component of endpoint stiffness between the vision and no vision conditions. This occurred despite similar levels of endpoint forces for both visual conditions in the NF. When the stiffness ellipses are examined for the individual subjects  = 6.767, p<0.001) indicating that the NF stiffness was not simply scaled up equally in the x- and y-axes. In the non-visual condition the xxK stiffness after adaptation to the DF was 2.30 times the size in the NF and for the yyK term it was 1.53 times the size. Similarly this increase was significantly larger in xxK than in yyK  = 4.982, p = 0.002). Therefore, the relative increase in the stiffness after adaptation to the DF occurred primarily in the x-axis for both the visual and non-visual conditions. This indicates that the stiffness had still been selectively increased in the direction of the instability of the force field.As observed in previous studies, stiffness ellipses in the DF were mainly increased towards the direction of instability without any changes in endpoint force. The selective increase in x-stiffness was seen under both visual feedback conditions which can be interpreted to mean that subjects adapted their endpoint impedance to attain stability in the DF in a similar manner whether or not online visual feedback was available. No significant differences were found between the two visual conditions for the ellipse orientation or shape. However there was a significant difference in the size of the stiffness ellipses between the two visual conditions. Examination of the stiffness matrices demonstrated that the no-visual condition was larger in both the xxK (the stiffness in the x-direction produced by a perturbation in the x-direction) increased by a greater percentage than yyK (the stiffness in the y-direction produced by a perturbation in the y-direction). This indicates that under both conditions the endpoint stiffness was increased primarily in the direction of the instability. Under the visual feedback condition subjects were able to reduce their handpath error more quickly. They also tended to make straighter movements when in either the NF or CF fields. In the same two fields, the visual movements also exhibited less variability than the non-visual trajectories but only towards the end of the movements.Eight subjects performed movements in three environments: a null field (NF), a stable force field (CF) and an unstable force field (DF), under two different visual conditions. Whether subjects were either presented with or without online visual feedback of their hand trajectory, they learned to compensate for both unstable and stable dynamics in order to make smooth movements to the target. Similar levels of endpoint force and endpoint stiffness for both visual feedback and non-visual feedback conditions were seen after adaptation to the two force fields. These results indicate that whether subjects were presented with visual feedback or not, the subjects adapted to the force fields in the same manner: learning the appropriate level of joint torques and endpoint stiffness for the environment. One distinct difference between the vision and no-vision condition was that the endpoint stiffness ellipse was slightly larger after adaptation to the DF with no visual feedback. However in both visual conditions Visual feedback of the trajectory is not required for adaptation to stable dynamics, nor is it required for adaptation to unstable dynamics. Subjects were able to adapt to the force fields both with and without visual feedback of their trajectory. Previous work has shown that subjects do not rely on visual feedback for learning stable dynamics One interesting finding with the speed of adaptation data was the analysis of the order effects. In particular when the subjects had previously learned the CF with either of the visual conditions, the speed of adaptation was increased for the second adaptation with the opposite visual condition. On the other hand, no such effect was found for adaptation to the unstable DF. When subjects had previously learned the field with one visual condition or not, the speed of adaptation was not affected. One possible interpretation of these results is that previous learning of unstable dynamics does not assist with the re-learning of the same dynamics. This is in contrast with the learning of stable dynamics, where the previous learning of similar dynamics can either assist or hinder the learning of the current dynamics depending on the similarity of the force fields While it is clear that the subjects could adapt to the novel dynamics both with and without visual feedback, the larger question is whether the adaptation occurs in the same way. It is possible to imagine that if different learning signals were used for adaptation with or without visual feedback, that the end control mechanism  could be different. We tested this by examining both the endpoint force and limb stiffness. If subjects used different methods of control with and without visual feedback, leading to different patterns of muscle activation, this would be seen both in the force trace and the endpoint stiffness. For example, if subjects modified the level of muscle co-activation, then this would have changed the shape and/or size of the endpoint stiffness ellipse, whereas a change in the reciprocal activation would have modified both the endpoint stiffness ellipse and the endpoint force. However, if these two measurements are similar both with and without vision, then it would indicate that subjects adapted in a similar style both with and without visual feedback. Previous work examining the adaptation to stable dynamics has not tested whether the adaptation occurred in the same way y-axis, the same pattern of endpoint force is used to accelerate the arm and produce the movement. In the x-axis, the endpoint force is maintained close to zero throughout the movement. Similarly, subjects both with and without vision produced roughly straight movements with no difference in the amount of linearity across the conditions. When the endpoint stiffness was examined there were some similarities and some differences in the endpoint stiffness between the two visual conditions. In both the visual and non-visual conditions, the relative increase in the stiffness after adaptation to the DF occurred primarily in the x-axis (increase in xxK was larger than the increase in yyK). This indicates that the stiffness had still been selectively increased in the direction of the force field. However the endpoint stiffness was larger when subjects were moving in the unstable environment with no visual feedback. Why would subjects tend to increase their endpoint stiffness in this unstable environment when no visual feedback was present? One possibility is that this arises from increased uncertainty about the environment. The brain needs to estimate the current state of the arm in order to compensate accurately for any small errors in the movement. However, the delayed and noisy sensory feedback and signal dependent noise in the generated motor command produce uncertainty in the brain's estimate of the current state After adaptation to the unstable DF dynamics, subjects again show similar levels of endpoint force, both with and without vision. In the This increased uncertainty about the current state would occur in both the stable and unstable environments without visual feedback, so why did the subjects only increase their limb stiffness in the absence of visual feedback  in the unstable environment? The answer to this is related to the difference between stable and unstable environments . In the x- and y-axes rather than only in the direction of the instability. One possible explanation is due to the problem with determining the appropriate orientation of the stiffness ellipse with respect to the hand. Depending on the position of the hand, the orientation of the hand, and therefore the orientation of the endpoint stiffness of the hand, relative to the environment would change. If the removal of visual feedback reduces the accuracy of the prediction of the future state of the subject's hand during movement, the exact orientation of the stiffness ellipse appropriate to counteract the instability in the environment may not be able to be predicted. The subjects may increase the stiffness in both the x- and y-axes in order to produce the appropriate level of stiffness for a variety of likely arm configurations. This possibility could be tested by adding one condition where the subjects only received visual feedback about the extent of their movements, comparable to the work of It is not clear, however, why the increase in stiffness in the no visual feedback condition  occurred in both the In a previous kinematic study, a statistically significant increase in hand-path curvature was found when subjects' vision was removed using a blindfold as compared to the case when visual feedback was available Previous studies have shown that visual feedback of hand position throughout the movement is used to correct the hand trajectory at relatively short delays In goal-directed movements under normal circumstances visual and proprioceptive feedback can be integrated by the CNS in order to localize the arm and the target and to track the execution of movement. It has been found that this multi-modal sensory integration is done so that each mode is optimally weighted and integrated in such a way as to minimize the effects of sensory noise in terms of direction-dependent finite precision and accuracy The sensorimotor system's dependence on somatosensory rather than visual information for the learning of dynamics may be unsurprising when considering the transformations required to determine the appropriate change in the feedforward motor command based on an error signal from each of these sensory systems


The basic causes of the radiation damage inflicted on macromolecular crystals during diffraction experiments are summarized, as well as the current state of research which attempts to understand and to mitigate it. Radiation damage inflicted during diffraction data collection in macromolecular crystallography has re-emerged in the last decade as a major experimental and computational challenge, as even for crystals held at 100 K it can result in severe data-quality degradation and the appearance in solved structures of artefacts which affect biological interpretations. Here, the observable symptoms and basic physical processes involved in radiation damage are described and the concept of absorbed dose as the basic metric against which to monitor the experimentally observed changes is outlined. Investigations into radiation damage in macromolecular crystallography are ongoing and the number of studies is rapidly increasing. The current literature on the subject is compiled as a resource for the interested researcher. Dose may also be quoted in terms of the non-SI unit rad . In MX, dose measurements are generally of the order of a million grays (1 MGy or 100 Mrad).The advent of highly intense wiggler and undulator beamlines fed from synchrotron sources has reintroduced the age-old problem of X-ray radiation damage in macromolecular crystallography (MX) even for crystals held at cryogenic temperatures (100 K). Unfortunately, such damage to macromolecular crystalline samples during the experiment is a problem that is inherent in using ionizing radiation to obtain diffraction patterns and has presented a challenge to MX since the beginning of the field. For room-temperature (RT) data collections, it often necessitates the use of many crystals to assemble a complete data set, because the crystalline order of the sample is damaged and decreases during the experiment and thus the diffracted intensity fades. The root cause of this damage is the energy lost by the beam in the crystal owing to either the total absorption or the inelastic scattering of a proportion of the X-rays as they pass through the crystal. The measure of this energy loss is the ‘dose’ measured per mass of the sample, given in SI units of grays  an undamaged fraction (A            1) which is entirely responsible for the remaining diffraction at high angles, (ii) a highly disordered fraction (A            2) which only contributes to the diffraction at low angles and (iii) a thoroughly disorganized or amorphous part [1 − (A            1 + A            2) = A            3] which no longer contributes to the single-crystal diffraction at all. From their plot of A            1, A            2 and A            3 against time derived from the seven successive sets of measurements and using their dose estimates it can be deduced that half of the crystal volume became amorphous after a dose of 0.59 MGy.The earliest investigation of radiation damage at RT in MX was carried out nearly 50 years ago by Blake & Phillips 1962 on a sea35S; Pechère et al., 1958Blake & Phillips 1962 also suget al., 1973et al., 1976et al., 2007A            1 and A            3 and thus the rate constant for transition from undamaged to amorphous was zero. Additionally, it was found necessary to include an intermediate dose-dependent stage labelled A            1′ between the undamaged and the damaged stages as shown in (2)I            0 of a strong reflection as the experiment proceeded and to discard the crystal once the intensity had dropped to 0.85I            0, or at the very worst 0.70I            0 if the particular crystals were in very short supply  that is usually more gentle than the capillary method historically used for RT collection; the fact that higher resolution data can more easily be obtained because the crystal order is preserved for longer; a lower background in the diffraction experiment as it is not necessary to enclose the crystal in a glass, quartz or plastic tube to prevent dehydration; that fewer crystals (and thus a lower quantity of protein) are required for a project; that crystals can be shipped ahead of time to the synchrotron (more or less) safely; and that crystals can be flash-cooled when in peak condition for future use before they start to degrade in the crystallization drop.These positive aspects of cryocooling commonly outweigh the disadvantages. The latter include the requirement for expensive cryostat cooling equipment, a frequent increase in crystal mosaic spread , the need to invest time for optimization of cryo-buffers and cooling protocols, and the fact that there are as yet no protocols that guarantee success, although progress is being made in this direction , an increase in both the scaling B factors for the data and the atomic B values of the refined structure, rotation of the molecule within the unit cell and often  an increase in mosaicity. Visible differences in the samples as the experiment proceeds, including colour changes, are also observed. On warming of the sample following irradiation, bubbles of gas, now proposed to be hydrogen  of a complete data set  after a dose D and I                     1 is the mean intensity of the first data set. Note that using I                     D/σD  normalized to the intensity I                     1/σ1 of the first data set is not a robust metric since the noise σD increases with dose and thus I                     D/σD reduces by an amount that more than represents the true loss of diffracting power.(i) R                     d, the pairwise R factor between identical and symmetry-related reflections occurring on different diffraction images, plotted against the difference in dose, ΔD, between the images on which the reflections were collected  B factor (B                     rel) has been found to be a robust measure of radiation damage at 100 K and to be linearly dependent on it  The isotropic et al., 2002(iv) The volume of the unit cell increases more or less linearly with dose and was originally thought to be a possible metric for judging the extent of radiation damage; however, systematic work  Although mosaicity commonly increases with dose, it is  not a reliable metric for quantization of radiation damage, since it does not behave in a reproducible or predictable manner.         Various metrics have been suggested and used for monitoring global damage, among which are the following.et al., 2000et al., 2002et al., 2005Of more direct relevance to the biological interpretation of structures than the global indicators detailed above is the fact that specific structural damage to particular covalent bonds is observed to occur in a reproducible order in many proteins  for a dose-rate increase from 4 × 103 to 40 × 103 Gy s−1 at flux densities of 4 × 1012 and 4 × 1013 photons s−1 mm−2, respectively.The global effects of radiation damage at 100 K are thought to be independent of dose rate up to the flux densities currently used , 7b) and 7c). It is worth noting that for a 100 µm thick protein crystal only 2% of the incident photons of a 12.4 keV (1 Å) X-ray beam will interact in any way with it .Radiation damage to the sample is a result of it absorbing photons from the beam by either the photoelectric effect  or Compton scattering . At the incident energies used for MX, the former effect has a much higher cross-section and dominates the absorption, accounting for over 90% of the energy deposited by the beam. Each photoelectron has enough energy to subsequently induce up to ∼500 further ionization events, which in turn can result in the formation of radical species in the crystal. In protein crystals, the presence of anything between 20% and 80% solvent means that the radiolysis of water and other components of the solvent is an important contributor to the creation of these species. Some of the energy deposited by the beam during these processes is converted into heat and induces a temperature rise in the sample. The diffracted photons are scattered elastically and thus do not contribute to the damage. These processes are illustrated diagrammatically in Figs. 7et al., 2002(i) Primary damage is the ionization of an atom owing to photoelectric absorption or Compton scattering. The primary photoelectron has a mean track length of a few micrometres . The secondary electrons gradually become thermalized  and chemical reactions between the radiation-induced moieties and the crystal components then become important.(ii) Secondary damage is that arising from the formation of up to 500 low-energy secondary electrons per primary absorption event, which are able to diffuse and induce further ionization and excitation events ((iii) Tertiary damage is defined as the effect on the crystal lattice and other mechanical consequences of the energy deposition in the crystal.         The usage of the terms ‘primary’, ‘secondary’ and ‘tertiary’ damage has become somewhat inconsistent in the literature and is largely a matter of semantics, but the definitions that will be adopted here are as follows. Damage can also be classified as direct, if the primary absorption event occurs at an atom in the protein molecule, or indirect, if the radiation is absorbed by the surrounding solvent and the reactive species formed subsequently interact with the protein. Energy deposition in the water in and around the crystal results in a cascade of reactions as shown below . In this case, translation of the crystal to a new position can be unsuccessful as a strategy for obtaining more data, since the damage can spread several tens of micrometres along the crystal from the irradiated position.An important experimental consideration is that at cryotemperatures the damage does not usually spread along the crystal. The crystal can thus be translated with respect to the beam so that fresh undamaged crystal can be irradiated at multiple positions. The exception is when heavy-atom clusters are present. The high absorption of these atoms can result in local heating of the crystal above 110 K, at which diffusion of With the advent of X-ray microbeams, the question arises as to how close sequential irradiations can be made while ensuring that ‘fresh’ material is in the beam, and there is ongoing systematic research to investigate this .4.3 unit cell would change the intensity of a 3 Å reflection by 15%  that can be tolerated as a function of the resolution limit of the data set (d            min): X = d            min/4  structure determinations, since by the time the second or third wavelength is collected, the cell and atomic structure can have changed such that the reflection intensities are significantly altered. This effect can obscure the anomalous signal required for structure solution. It has been calculated that a 0.5% change in all three dimensions of a 100 ÅI4 can be indexed with the b axis pointing in either direction, so that when data are merged care must be taken that each section is indexed in the same convention.Secondly, the radiation-sensitivity of some crystals at 100 K means that it is not possible to collect a complete data set from a single crystal and data must be merged from several (or many) of them to measure all the unique reflection intensities. Although this was routinely the case when data were collected at RT, most crystallographers have become accustomed to being able to measure all unique reflections from just one cryocooled crystal. Use of multiple crystals to assemble a complete data set in general increases the errors arising from non-isomorphism, thereby potentially reducing the ease of structure solution as well as increasing the mounting/dismounting time burden. Even using a robot for this operation can be slow and in fact is sometimes the most time-consuming part of the experiment. It can also present some pitfalls during processing. For instance, space group et al., 2005e.g. Takeda et al., 2004Finally, the radiation-damage-induced structural changes can affect the apparent biological properties of the macromolecule under study. Enzyme mechanisms can involve redox-susceptible residues, so special care is required when interpreting structures that may have been modified by X-ray damage during the data collection. For instance, irradiation can change the oxidation state of metal ions in structural/active sites from that in their native state  in the irradiated volume. The fact that the amount of damage at 100 K is indeed proportional to the absorption coefficient and thus to the dose has been shown in elegant experiments by Kmetko et al.  during two-dimensional diffraction experiments in electron microscopy, Henderson  was found to be 43 MGy, although the recommended maximum dose was only 30 MGy in order to avoid compromising the biological information extracted from deduced structures. This dose limit corresponded to a drop in diffraction intensity to 70% (D            0.7) of the initial value . This issue is described in detail later in this volume  of its original intensity. A crystal might not survive until the limit is reached  for hen egg-white lysozyme crystals  has been shown to be improved by factors of ∼2 and ∼9 by the addition of the scavengers ascorbate and 1,4-benzoquinone, respectively , can be modified by the addition of scavenger molecules to become a zeroth-order dependence (where the rate of decay is a constant). This effect is not yet completely understood. At RT, the dose tolerance of HEWL crystals (as measured by the change in i.e. the number of each atom type in the unit cell) is required so that absorption coefficients can be computed, as well as detailed information about the incident beam . For MX, this can be conveniently carried out by means of the program RADDOSE  a crystal. There are two challenges for researchers seeking to understand and trying to mitigate radiation damage. The first is to truly isolate the variable to be tested and to only change one experimental condition at a time so that definite conclusions can be reached. The second is to use a reliable metric(s) of radiation damage so that the effect of protocol modifications can be properly assessed. In addition, to reach a statistically significant result the same experiments must be repeated and reproduced on several crystals of the same protein and ideally extended to check the validity of the results in a more general way by conducting the same tests for a number of different proteins.vice versa.Over the last 10 years there has been an extensive search for reliable metrics of global and structurally specific radiation damage. Since structural changes occur even before degradation of diffraction quality is apparent, intensity loss cannot be used as a yardstick with which to judge damage to specific amino acids, which is only obvious when the electron-density maps have been calculated once enough data have been collected. This can be understood because the diffraction loss occurs in reciprocal space and the specific damage in real space, and one point in real space contributes to all reflections in reciprocal space and etc.), solvent content, solvent composition, crystal size and surface-to-volume ratio, the amount of residual liquid around the crystal prior to flash-cooling, the choice and concentration of cryoprotectant agent, the time spent in the cryobuffer prior to cooling, the flash-cooling method (stream or liquid), the cryogen used to flash-cool, the amount of crystal manipulation, the local humidity and the speed of the experimenter when flash-cooling from the cryobuffer drop.(i) The crystal in the cryo-loop: heavy-atom content  The X-ray beam: the flux density, the energy (wavelength), the beam size compared with the crystal size, the dose and the dose rate.2 or He).(iii) The cryostat: the cold gas flow rate, the temperature and the cryogen ?What is the effect of temperature  Crystal related.What is the effect of changing the incident wavelength?Is it beneficial to change/regulate the dose/dose-rate regime?What is the effect of the beam size compared with the crystal size?Does the beam heat the crystal?                  (ii) X-ray beam related.Development of convenient flux calibration of beamlines.Development of on-line and off-line spectroscopy .Studying the effect on the success of MAD/SAD phasing.Development of RIP/RIPAS.Application/effect of radiation damage to/on the study of biological mechanisms.Phase transitions and/or radiation-induced changes with temperature-controlled cryocrystallography to study macromolecular function.Software developments.Finding strategies to minimize radiation damage in data collections.Extending the understanding of radiation damage in RT data collections.Studying RNA/DNA damage.                  (iii) Method developments and applications.         Systematic experiments to address the dependence of the rate of radiation damage on all these factors would take many years and be very labour-intensive in terms of data collection and processing, as well as requiring many hours of synchrotron beamtime. However, some of these variables have been investigated and studies can be broadly categorized as follows.Journal of Synchrotron Radiation, which each contain eight or more research papers presented at the Second, Third, Fourth and Fifth International Workshops on X-ray Radiation Damage to Biological Crystalline Samples . These each have a brief introduction placing the contributions into the wider context of research in the field  or in a soaking solution for heavy-atom phasing, since these heavy atoms can contribute a lot to the dose owing to their high absorption but do not provide useful phasing information; (ii) sacrificing a crystal  to obtain a data set where the aim is to assess the radiation-sensitivity so that a suitable data-collection protocol can be designed; (iii) matching the beam size to the crystal size; (iv) if possible using a beam with a top-hat profile  so that the crystal does not suffer differential radiation damage across its irradiated volume; (v) using BEST to optimize the data-collection strategy taking radiation damage into account  backsoaking of crystals to remove any nonspecifically bound heavy atoms in the mother liquor (Above all, experimenters should make themselves aware of the parameters known to affect the rate of radiation damage, so that intelligent choices/compromises can be made.7.Since systematic research into MX radiation damage at 100 K began in earnest in the late 1990s, significant progress has been made in our knowledge and understanding of the phenomenon and much anecdotal evidence has been replaced by solid experimental results. We understand better how to perform investigations to identify suitable metrics and the importance of routinely measuring the X-ray flux so that the absorbed dose can be calculated. The research has also prompted some exciting new approaches such as RIP, UV-RIP/RIPAS, ‘time-resolved’ cryocrystallography and on-line spectroscopy. However, there are still many areas where systematic investigations are required to improve our understanding of the radiation chemistry within an irradiated protein crystal held at either RT or at various cryotemperatures so that better strategies for minimizing damage can be developed.The most useful contribution to be made by MX radiation-damage research is in identifying concrete experimental protocols for everyday use on synchrotron beamlines so that researchers can ensure that they obtain the maximum possible high-quality data from their crystals. This would firstly facilitate structure solution and secondly avoid compromising the biological information extracted from the structure once obtained.


Zeta potential is the key parameter that controls electrostatic interactions in particle dispersions. Laser Doppler electrophoresis is an accepted method for the measurement of particle electrophoretic mobility and hence zeta potential of dispersions of colloidal size materials. Traditionally, samples measured by this technique have to be optically transparent. Therefore, depending upon the size and optical properties of the particles, many samples will be too concentrated and will require dilution. The ability to measure samples at or close to their neat concentration would be desirable as it would minimize any changes in the zeta potential of the sample owing to dilution. However, the ability to measure turbid samples using light-scattering techniques presents a number of challenges. This paper discusses electrophoretic mobility measurements made on turbid samples at high concentration using a novel cell with reduced path length. Results are presented on two different sample types, titanium dioxide and a polyurethane dispersion, as a function of sample concentration. For both of the sample types studied, the electrophoretic mobility results show a gradual decrease as the sample concentration increases and the possible reasons for these observations are discussed. Further, a comparison of the data against theoretical models is presented and discussed. Conclusions and recommendations are made from the zeta potential values obtained at high concentrations. It can be used to optimize the formulations of suspensions and emulsions and aid in predicting long-term stability . Most coΔf is equal toν is the particle velocity, λ is the laser wavelength and θ is the scattering angle. The measured electrophoretic mobility (UE) is converted into zeta potential (ζ) through Henry’s equation,ε is the dielectric constant of the dispersant, F(κa) is the Henry function and η is the viscosity (F(κa)=1.5 is known as the Smoluchowski equation (κa is large (approx. 100) and the double layer is thin in comparison with the particle radius. This limit is relevant to some but not all of the sample measurements presented in this paper. Hence the data are given in the form of mobility, and the more complex formulations needed to deal with the derivation of zeta potential, where the particle is not isolated and the zeta potential not small, are addressed in a later section.The zeta potential of colloidal dispersions is routinely measured using the technique of micro-electrophoresis  measurements, the laser beam has to penetrate the sample for scattered light at a forward angle of 13detected . This opOne way to measure the electrophoretic mobility of samples at higher concentration is to reduce the path length of the cell, which will increase the transmission of the laser through the sample. Even though such a cell will allow electrophoretic mobility measurements to be made on concentrated samples, the conversion to zeta potential, and subsequent interpretation of what the results obtained mean, is not straightforward.This paper summarizes measurements made on different types of samples at high concentration and turbidity and discusses the concentration dependence on the zeta potential values obtained.2.°C on a Zetasizer Nano ZS  fitted with a high-concentration zeta potential cell (ZEN1010).All measurements reported in this paper were made at a temperature of 25−5 to 7.092×10−4). Dilution in an indifferent electrolyte, such as NaCl, should ensure that any changes in the zeta potential values obtained were not due to conductivity differences. (ii) A polyurethane dispersion at a volume fraction of 0.4 was kindly provided by Baxenden, a Chemtura company. Various concentrations were prepared in 5 per cent v/v triethylamine to maintain ionic strength and ensured that any differences in the measured electrophoretic mobilities were not due to changes in conductivity.Two samples were measured in this study. (i) A titanium dioxide (anatase) sample was prepared in 10 mM NaCl at a range of volume fractions .For all measurements, a field of 40 V was applied across an electrode spacing of 16 mm. Five repeat measurements on each sample were made to check the repeatability of the results obtained. All measured electrophoretic mobilities were converted into zeta potential using Smoluchowski’s formula (°. The intensity-averaged particle diameters and the polydispersity index (PDI) values (an estimate of the distribution width) were calculated from the cumulants analysis as defined in ISO13321 (Size characterization of the samples was made by dynamic light-scattering (DLS) measurements using the Zetasizer Nano ZS, which uses a 4 mW He–Ne laser operating at a wavelength of 633 nm and a detection angle of 173ISO13321 . The intISO13321 .3.(a)2 sample prepared in 10 mM NaCl. The table also includes the measured conductivity values for these samples and illustrates that the presence of 10 mM NaCl maintains a consistent ionic strength across the concentration range. The standard deviation values are obtained from the five repeat measurements performed on each sample. The electrophoretic mobility values are plotted as a function of volume fraction in (b)The intensity particle size distributions obtained from the three repeat measurements are over plotted in 2 V s−1×10−8) measured as a function of volume fraction for the polyurethane dispersions dispersed in 5 per cent v/v triethylamine (apart from the neat sample with a volume fraction of 0.4).4.(a)The optical configuration used in a typical laser Doppler instrument such as the Zetasizer Nano means that the laser beam has to penetrate the sample for scattered light to be detected at a forward angle. Therefore, in general, samples for electrophoretic mobility measurements have to be optically clear. As discussed above, the maximum concentration at which the electrophoretic mobility of a sample can be measured will depend upon a number of factors such as particle size, polydispersity of the particles and optical properties. The optical properties (refractive index=2.4) and particle size distribution  of titan−5 to 4.137×10−4 volume fraction range have a mean value of −48.9 mV with a standard deviation of 2.5 mV. However, as the concentration increases above a volume fraction of 4.137×10−4, the mobility values become less negative, i.e. the magnitude of mobility decreases.The high-concentration zeta potential cell, with its reduced path length, allows electrophoretic mobility measurements to be made on samples which are turbid and at high concentration. However, the influence of concentration on the mobility results obtained needs to be studied and understood. In the results summarized in −4), the viscosity was determined to be that of water (0.89 mPa.s at 25°C).There are various possibilities for the concentration dependence in the mobilities obtained. Dilution may have led to a change in sample equilibrium. This is unlikely as the samples were prepared in a non-specific electrolyte of sufficient concentration (10 mM NaCl) to maintain consistent conductivity for each dilution . The effThe most probable explanation is that the increase in sample concentration leads to higher turbidity, resulting in obscuration of light transmission. Therefore, the scattered light being detected originates from particles close to the wall, rather than the centre, of the cell where they have reduced mobility owing to the electric field being lower than assumed.(b)The polyurethane dispersion used in this study is ideally suited for looking at the influence of concentration on the electrophoretic mobility and hence zeta potential. Measurements can be made even at the neat concentration of 0.4 volume fraction with the high-concentration zeta potential cell owing to the particle size of around 50 nm and the low polydispersity index values.2 V s−1×10−8). However, at concentrations above a volume fraction of 0.1, there is a gradual decrease in the electrophoretic mobilities measured. This is not unexpected as the viscosity of the sample increases significantly with concentration.F(κa) of 1.5 (i.e. Smoluchowski’s approximation) and the dielectric constant of water were used. The viscosities of the samples measured on the SV-10 vibroviscometer were used to calculate the zeta potential values. The zeta potential results obtained from these calculations are plotted as a function of sample concentration in Henry’s equation relates the electrophoretic mobility to zeta potential through Henry’s function, the dielectric constant of the dispersant and the sample viscosity. In this study, an The zeta potential values at the higher sample concentrations are not realistic, indicating that the viscosity values obtained from a simple viscometer such as the SV-10 vibroviscometer cannot be used. The macroscopic viscosity measured by the viscometer is probably not representative of the local viscosity which the particles experience as they undergo electrophoresis. These polyurethane dispersions are complex samples probably exhibiting non-Newtonian behaviour and, therefore, it is uncertain which viscosity value is the most appropriate to use for the conversion of the electrophoretic mobility data into zeta potential values.If the viscosity is assumed to be that of water for all volume fractions of the sample, the zeta potential values obtained are more realistic . However(c)There are relatively few references which discuss concentration dependence on electrophoretic mobility and even fewer exist that compare the predictions of these models with experimental evidence. (d)μ at volume fraction φ and the mobility at infinite dilution, μ0, is given byThe Anderson relationship between the electrophoretic mobility tionship . This re(e)Ohshima extends the work of Levine and Neale to a more general case to include the work of Kozak & Davis a,b. Theiκa=2.5 curve above a normalized volume fraction of 0.1, the κa value for this sample is computed as approximately 5. From κa=5 curve occurs at about a volume fraction of 0.07 in much the same region as the Anderson model. It is far beyond the scope of this paper to discuss the comparison of the two models with the experimental data in great detail. For now, the curiosity that they both fail at the same concentration despite the obvious differences in approach is noted here. Effects not modelled in either case include the viscoelectric effect and changes in the electrochemical potential of the ions in the system owing to the higher volume fraction, although it is acknowledged that the similar point of failure may simply be coincidental. A more rigorous study will be reported at a later date.Despite the apparent fit to the It is important to note, however, that, despite the discrepancies between data and model above volume fractions of 0.1, the small particle size and low refractive index of this sample mean that the measured change in mobility can, in this case, be taken as real and not an artefact of the measurement technique.5.The results presented in this paper confirm that high-concentration zeta potential measurements of samples up to volume fractions of 0.4 can be made using light-scattering techniques when a suitable optical configuration, such as a cell of reduced path length, is used. The maximum measurable concentration of a sample will depend on the mean particle size, the polydispersity of particles and the optical properties.The results obtained often show a concentration dependence and such trends need to be carefully understood and interpreted. For both of the sample types discussed in this paper, the electrophoretic mobility results show a gradual decrease as the sample concentration increases. However, this trending in the electrophoretic mobility data occurs for different reasons for each sample. For the titanium dioxide measurements, the reduction is an optical effect owing to the increasing turbidity of the sample, which results in an increase in the obscuration of transmitted light. For the polyurethane dispersions, the trend is not due to an optical effect as the intensity of scattered light being detected is still very high even at volume fractions of 0.4.The conversion of the measured electrophoretic mobility values into zeta potential using Henry’s equation requires the viscosity of the sample to be entered. In a complex sample such as the polyurethane dispersion used in this study, it is difficult to know what the most appropriate viscosity value to use is. The results summarized in From the results obtained in this study, it can be concluded that zeta potential values obtained at high concentrations should be used in a relative, not absolute, sense. If the true zeta potential value of a sample is required, measurements should be performed at a range of concentrations to identify a region where the zeta potential results obtained are independent of concentration. For these experiments, the dilution protocol is pivotal to ensuring that sample conditions are kept constant.Two models of mobility were tested against the experimental data collected and shown to provide a good estimate of the change in mobility owing to high concentration below normalized volume fractions of about 0.1.


Phylogenetic analysis is becoming an increasingly important tool for biological research. Applications include epidemiological studies, drug development, and evolutionary analysis. Phylogenetic search is a known NP-Hard problem. The size of the data sets which can be analyzed is limited by the exponential growth in the number of trees that must be considered as the problem size increases. A better understanding of the problem space could lead to better methods, which in turn could lead to the feasible analysis of more data sets. We present a definition of phylogenetic tree space and a visualization of this space that shows significant exploitable structure. This structure can be used to develop search methods capable of handling much larger data sets. Phylogenetic analysis has become an integral part of many biological research programmes. These include such diverse areas as human epidemiology ,2, viralPhylogenetic search is a difficult problem. When parsimony is used as the optimality criterion the problem is known to be NP-complete . The seaA variety of heuristic search methods have been used to find optimal trees within a tree space. The most common method is to search tree space using tree rearrangements -11. OtheGreater understanding of the problem space may allow more sophisticated search techniques to be applied, with a consequent improvement in the effectiveness of the search. One technique that can be used to better understand the space of phylogenetic search, and the behavior of search algorithms within this space, is visualization. This includes two separate activities: first, defining the search space of phylogenetic trees, or tree space, and second, developing methods to display tree space in a way that is exploitable in search techniques.This visualization must have the following properties to be useful.• Each tree should map to a single deterministic position. Otherwise the method is restricted to post-processing, and cannot be used to guide a search.• Distance between trees should be easy to calculate. If it is not, the visualization will not be able to be used in real time to guide a search.• The visualization should reveal exploitable structure. This is important because if a visualization shows no structure it provides no guidance for a search.• This mapping should be reversible, meaning that there should be a method of turning a position into a tree. This is necessary, as to be useful in searching it must be possible to quickly find trees in the space suggested by the visualization.This work presents an elegant linear projection of trees. This projection can be computed much faster than current alternatives and is better at preserving structural continuity between trees after the projection. Furthermore this projection is deterministic, allowing it to be used as an inline rather than a post-process analysis. This property coupled with the structural preservation allows the consideration of novel search strategies in the new projected space.The Results and Discussion section presents a definition of tree space and an elegant projection of that space that has all four of these desirable properties. This projection is then used to visualize the tree space and expose structure that can be exploited to guide the searches of common, but computationally expensive, methods.Tree space consists of all of the possible phylogenetic trees for a given set of taxa and their relationships with each other. This space is the domain of whatever search strategy is employed. Previous search strategies have not explicitly defined this domain, and the tree space that implicitly arises from these strategies is very cumbersome to work with. Tree spaces have also been explicitly defined without designing algorithms to take advantage of these spaces. This is primarily due to a lack of exploitable structure in these explicitly defined tree spaces. Figure The most common tree spaces used in phylogenetic search are the spaces implicitly defined by the subtree transfer operations, such as Tree Bisection and Reconnection (TBR) or Subtree Prune and Regraft (SPR), used during the search. These operations in turn induce distances between trees . These tet al. . Using the Gram-Schmidt method these vectors were all made to be orthogonal to each other. Finally they were each modified to make their magnitudes equal to the magnitude of the first vector.Definition 5. h is a function with the following properties:d <d' is h(x) = x mod d.Such a function is easily constructed. One such function when Definition 6. X' and Y' are two vectors constructed from X, Y, and h as follows:Definition 7. The frequency with which h maps any number j onto a given number i isfi has the following bounds:Note that Definition 8. ξ, the quality of the function h, is a measure of how evenly the elements of X' and Y' are mapped by h onto X and Y.fi, ξ has the following bounds:Note that due to the bounds on all Theorem 9. X and Y, two arbitrarily larger vectors X' and Y' can be constructed such that they are linearly independent.Given two orthogonal vectors Proof. As X and Y are orthogonal, they are also linearly independent. That is to say:Thus all equations of the formcan be rewritten asIn this fashionX' and Y' are linearly independent.From which it is clear that Theorem 10. X and Y, two arbitrarily larger vectors X' and Y' can be constructed such that they are orthogonal within a given bound.Given two orthogonal vectors Proof. Using Definition 7 the inner product ⟨X'|Y'⟩ can be written in terms of X and Y.X and Y are orthogonal, their inner product ⟨X|Y⟩ = 0; therefore eitherAs and clearlyorX' and Y' are orthogonal. Even so there are bounds on ⟨X'|Y'⟩. The largest possible magnitude of ⟨X'|Y'⟩ occurs when h maps each member of {ℕ <d} to one member of {ℕ <d'} with the exception of one element of {ℕ <d} which maps to the remaining elements of {ℕ <d'}. Furthermore, that sole exception corresponds with the largest magnitude of XiYi. In this case the inner product is given byIn this second case it may not be true that θ between X' and Y' is given byThe angle X'|Y'⟩, and the bounds on the magnitudes of X' and Y' from Theorem 11Applying the bound on ⟨X and Y are arbitrary but constant expressions, note thatAs Therefore as the number of taxa increases the vectors in question approach orthogonality.Theorem 11. X and Y, two arbitrarily larger vectors X' and Y' can be constructed such that they are also of equal magnitude within a given bound.Given two vectors of equal magnitude Proof. As X and Y are of equal magnitude it is the case thatX' is bounded above byThe magnitude of h is in {ℕ <d} every element of X is also an element of X'. ThereforeAs the range of Y' is bounded in the same fashion. The ratio of the two magnitudes is bounded as followsThe magnitude of ξ = 0 thenAdditionally, if and the two vectors have equal magnitudeThe authors declare that they have no competing interests.KS developed and implemented the methods. QS and MC both acted in an advisory role. All authors read and approved the final manuscript.


We present a simple construction method for Feller processes and a framework for the generation of sample paths of Feller processes. The construction is based on state space dependent mixing of Lévy processes. Brownian Motion is one of the most frequently used continuous time Markov processes in applications. In recent years also Lévy processes, of which Brownian Motion is a special case, have become increasingly popular. Lévy processes are spatially homogeneous, but empirical data often suggest the use of spatially inhomogeneous processes. Thus it seems necessary to go to the next level of generalization: Feller processes. These include Lévy processes and in particular Brownian motion as special cases but allow spatial inhomogeneities. Many properties of Feller processes are known, but proving the very existence is, in general, very technical. Moreover, an applicable framework for the generation of sample paths of a Feller process was missing. We explain, with practitioners in mind, how to overcome both of these obstacles. In particular our simulation technique allows to apply Monte Carlo methods to Feller processes. The paper is written especially for practitioners and applied scientists. It is based on two recent papers in stochastic analysis The main part of the paper contains a simple existence result for Feller processes and a description of the general simulation scheme. These results will be followed by several examples.The source code for the simulations can be found as supporting information .Brownian motion and more general Lévy processes are used as models in many areas: For example in medicine to model the spreading of diseases In hydrology stable processes are used as models for the movement of particles in contaminated ground water. It has been shown that state space dependent models provide a better fit to empirical data In geology also stable processes are used in models for the temperature change. Based on ice-core data the temperatures in the last-glacial and Holocene periods are recorded. Statistical analysis showed that the temperature change in the last-glacial periods is stable with index 1.75 and in the Holocene periods it is Gaussian, i.e. stable with index 2  price-dependent, i.e. the increment of the process shall depend on the current price. This procedure is applicable to every class of Lévy processes, but the existence has to be shown for each class separately In mathematical finance the idea of extending Lévy processes to Lévy-like Feller processes was first introduced in Thus there is plenty of evidence that Feller processes can be used as suitable models for real-world phenomena.Up to now general Feller processes were not very popular in applications. This might be due to the fact that the existence and construction of Feller processes is a major problem. There are many approaches: Using the Hille-Yosida theorem and Kolmogorov's construction Our construction will not yield processes as general as the previous ones, but it will still provide a rich class of examples. In fact the presented method is just a simple consequence of a recent result on the solutions to certain stochastic differential equations Furthermore each of the above mentioned methods also provides an approximation to the constructed Feller process. Most of them are not usable for simulations or work only under technical conditions. Also further general approximation schemes exist, for example the Markov chain approximation in In contrast to these we derived in Lévy process and Feller process are sometimes used for different objects. Thus we will clarify our notion by giving precise definitions and mentioning some of the common uses of these terms.Within different fields the terms A stochastic process is a family of random variables indexed by a time parameter Although this will not appear explicitly in the sequel, a process will always be equipped with its so-called natural filtration, which is a formal way of taking into account all the information related to the history of the process. Technically the filtration, which is an increasing family of sigma fields indexed by time, is important since a change from the natural filtration to another filtration might alter the properties of the process dramatically.Lévy process starting in A - independent increments: The random variables - stationary increments: - càdlàg paths: Almost every sample path is a right continuous function with left limits.For equivalent definitions and a comprehensive mathematical treatment of Lévy processes and their properties see Lévy flight often refers to a process which is a continuous time random walk (CTRW) with spatial increments from a one-sided or two-sided stable distribution . In our notion the processes associated with these increments are Lévy processes which are called stable subordinator and stable process, respectively.Note that the term A Lévy process The most popular Lévy process is Brownian motion  that the process we want to model behaves like a Lévy process If the sets Let Now set for Note that the theorem extends to any finite number of Lévy processes To avoid pathological cases one should assume Thus if one knows how to simulate increments of the Given a Feller process Select a starting point The first point of the sample path is current position of the sample path).Draw a random number x  at time t .The next point of the sample path is Repeat 3. and 4. until The simulated path is an approximation of the sample path of the Feller process, in the sense that for The reader familiar with the Euler scheme for Brownian or Lévy-driven stochastic differential equations (SDEs) will note that the approximation looks like an Euler scheme for an SDE. In fact it is an Euler scheme, but the corresponding SDE does not have such a nice form as for example the Lévy-driven SDEs discussed in We will now present some examples of Feller processes together with simulations of their sample paths. The first example will show the generality of the mixture approach, the remaining examples are special cases for which the existence has been shown by different techniques.All simulations are done with the software package R To show the range of possibilities which are covered by the mixture approach we construct a process which behaves likeFor this we just define a family of Lévy processes by the family of characteristic exponents These functions are Lipschitz continuous and thus a corresponding Feller process exists. A Lévy process If we now define a function A corresponding Feller process exists and is unique if the function yellow the state becomes, the more the process behaves like Brownian motion.The characteristic function of a normal inverse Gaussian process Note that the mean reversion is not introduced by using simply a drift which drags the process back to the origin. It is the choice of The characteristic function of a Meixner process A family of Meixner processes which corresponds to a Feller process can be constructed by substituting the parameters moves with bigger steps around the origin, to be precise: the Meixner distribution has semiheavy tails The chosen functions satisfy the existence conditions from above. Furthermore the function Using the presented mixture approach one can easily construct Feller models based on given Lévy models. In these cases the existence of the process is granted.Furthermore the presented approximation is a very intuitive way to generate the sample path of a Feller processes. Obviously the method requires that one can simulate the increments of the corresponding Lévy processes. But for Lévy processes used in applications, especially together with Monte Carlo techniques, this poses no new restriction.Thus all necessary tools are available to use Feller processes as models for a wide range of applications.The simulations where done in R Appendix S1Source code of the figures.(TXT)Click here for additional data file.


We report a family with the Li-Fraumeni syndrome (LFS) in whom we have been unable to detect a mutation in the coding sequence of the p53 gene. Analysis of linkage to three polymorphic markers within p53 enabled direct involvement of p53 to be excluded. This is the first example of a LFS family in whom exclusion of p53 has been possible. Four affected members of the family with sarcoma or premenopausal breast cancer showed increased expression of p53 protein in their normal tissues as detected by immunohistochemistry. It therefore appears that the LFS phenotype has been conferred by an aberrant gene, showing a dominant pattern of inheritance, which may be acting to compromise normal p53 function rather than by a mutation in p53 itself. In order to try to determine the chromosomal location of this putative gene, we have carried out studies of linkage to candidate loci. By these means we have excluded involvement of Rb1 and BRCA1 on chromosomes 13q and 17q respectively. The MDM2 oncogene on chromosome 12q was considered to be the prime candidate as MDM2 is amplified in sarcomas and the MDM2 product binds to p53. Furthermore, p53 mutation and amplification of MDM2 have been shown to be mutually exclusive events in tumour development. Linkage analysis to two polymorphic markers within MDM2 yielded a three-point LOD score of -5.4 at a recombination fraction theta equal to zero. Therefore MDM2 could be excluded. It is possible that the gene which is responsible for cancer susceptibility in this family, possibly via interaction with p53, will be important in the histogenesis of breast cancer in general. We are now carrying out further studies to locate and identify this gene.


Fluorescent and luminescent gene reporters allow us to dynamically quantify changes in molecular species concentration over time on the single cell level. The mathematical modeling of their interaction through multivariate dynamical models requires the deveopment of effective statistical methods to calibrate such models against available data. Given the prevalence of stochasticity and noise in biochemical systems inference for stochastic models is of special interest. In this paper we present a simple and computationally efficient algorithm for the estimation of biochemical kinetic parameters from gene reporter data.We use the linear noise approximation to model biochemical reactions through a stochastic dynamic model which essentially approximates a diffusion model by an ordinary differential equation model with an appropriately defined noise process. An explicit formula for the likelihood function can be derived allowing for computationally efficient parameter estimation. The proposed algorithm is embedded in a Bayesian framework and inference is performed using Markov chain Monte Carlo.The major advantage of the method is that in contrast to the more established diffusion approximation based methods the computationally costly methods of data augmentation are not necessary. Our approach also allows for unobserved variables and measurement error. The application of the method to both simulated and experimental data shows that the proposed methodology provides a useful alternative to diffusion approximation based methods. The estimation of parameters in biokinetic models from experimental data is an important problem in Systems Biology. In general the aim is to calibrate the model so as to reproduce experimental results in the best possible way. The solution of this task plays a key role in interpreting experimental data in the context of dynamic mathematical models and hence in understanding the dynamics and control of complex intracellular chemical networks and the construction of synthetic regulatory circuits . Among bSingle cell imaging techniques have revealed the stochastic nature of biochemical reactions (see  for revi1. The macroscopic rate equation (MRE) approach which describes the thermodynamic limit of the system with ordinary differential equations and does not take into account random fluctuations due to the stochasticity of the reactions.2. The diffusion approximation (DA) which provides stochastic differential equation (SDE) models where the stochastic perturbation is introduced by a state dependant Gaussian noise.3. The linear noise approximation (LNA) which can be seen as a combination because it incorporates the deterministic MRE as a model of the macroscopic system and the SDEs to approximatively describe the fluctuations around the deterministic state.Statistical methods based on the MRE have been most widely studied ,13-15. TThe chemical master equation (CME) is the primary tool to model the stochastic behaviour of a reacting chemical system. It describes the evolution of the joint probability distribution of the number of different molecular species in a spatially homogeneous, well stirred and thermally equilibrated chemical system .N chemical species inside a volume Ω and let X = T denote the number and x = X/Ω the concentrations of molecules. The stoichiometry matrix S = {Sij}i = 1,2...N; j = 1,2...R describes changes in the population sizes due to R different chemical events, where each Sij describes the change in the number of molecules of type i from Xi to Xi + Sij caused by an event of type j. The probability that an event of type j occurs in the time interval [t, t + dt) equals x, Ω, t)Ωdt. The functions x, Ω, t) are called mesoscopic transition rates. This specification leads to a Poisson birth and death process where the probability h that the system is in the state X at time t is described by the CME [Even though these assumptions are not necessarily satisfied in living organisms the CME is commonly regarded as the most realistic model of biochemical reactions inside living cells. Consider a general system of  the CME  which islet X = X,..., XNTϕi = limX→∞ Ω→∞, Xi/Ω, φ = T and where Including also the second order terms of this expansion produces the LNAφ as solution of the MRE in (1) and a stochastic process ξ described by an Itô diffusion equationwhich decomposes the state of the system into a deterministic part W(t) denotes R dimensional Wiener process, fi = fi(φ)  = The LNA provides solutions that are numerically or analytically tractable because the MRE in (1) can be solved numerically and the linear SDE in (3) for an initial condition the form s) is the fundamental matrix of the non-autonomous system of ODEswhere the integral is in the Itô sense and ξ are Gaussian [The Itô integral of a deterministic function is a Gaussian random variable , therefoGaussian   is the normal density with mean μi-1 and covariance matrix Ξi-1 specified bywhere Θ denotes a vector of all model parameters, x are normalIt follows from (2) and (6) that the transition densities of The properties of the normal distribution allow us to construct a convenient inference framework that is reminiscent of the Kalman filtering methodology (see e.g. ).xt into those components yt that are observed and those zt that are unobserved.It is rarely possible to observe the time evolution of all molecular components participating in the system of interest . Therefox, y and z, respectively, at times t0,..., tn. Here and throughout the paper we use the same letter to denote the stochastic process and its realization.Let φ(t0) is parameterized as an element of Θ. Given the Markov property of the process x the augmented likelihood P(Our aim is to estimate the vector of unknown parameters Θ from a sequence of measurements where φ(·|φ(t0),..., φ(tn), φ(t0),..., φ(tn)) and covariance matrix P and ϵ, i.e. Fluorescent reporter data are usually assumed to be proportional to the number of fluorescent molecules  and measP(Θ|Since for given data relation We use the standard Metropolis-Hastings (MH) algorithm  to samplIn order to study the use of the LNA method for inference we have selected four examples which are related to commonly used quantitative experimental techniques such as measurements based on reporter gene constructs and reporter assays based on Polymerase Chain Reaction . For expository reasons, all case studies consider a model of single gene expression.Although gene expression involves various biochemical reactions it is essentially modeled in terms of only three biochemical species  and four reaction channels  -33. The x =  denote concentrations of mRNA and protein, respectively. For the reaction rateswhere rows correspond to molecular species and columns to reaction channels. Let we can derive the following macroscopic rate equationskR(t) is time-dependent, reflecting changes in the regulatory environment of the gene such as the availability of transcription factors or chromatin structure.For the general case it is assumed that the transcription rate Using 14), (15) and (16) in (3) we obtain the following SDEs describing the deviation from the macroscopic state  asimple model of single gene expression.We will refer to the model in (16) and (17) as the kR(t) now describes the maximum rate of transcription, H is a dissociation constant and nH is a Hill coefficient.In order to test our method on a nonlinear system we will also consider the case of an autoregulated network where the transcription rate of the gene is a function of the concentration of the protein that the gene codes for and where the protein is a transcription factor that inhibits the production of its own mRNA. This is parameterized by a Hill function  where kRThus, the nonlinear autoregulatory model the system is described by the MREand the SDEsautoregulatory model of single gene expression. The two models constitute the basis of our inference studies below.where To test the algorithm we first use the simple model of single gene expression. We generate data according to the stoichiometry matrix (14) and rates (15) using Stochastic Simulation Algorithm (SSA)  and sampλ is an unknown proportionality constant and where t ≤ b3 as a result of being induced by an environmental stimulus and for t > b3 decreases towards a baseline level b4.This form of transcription corresponds to an experiment, where transcription increases for t0 (t0 <<b3) the system is in a stationary state. Therefore, the initial condition of the MRE is a function of unknown parameters (ϕR(t0), ϕP(t0)) = .We assume that at time To ensure identifiability of all model parameters we assume that informative prior distributions for both degradation rates are available. Priors for all other parameters were specified to be non-informative. To infer the vector of unknown parameterswe sample from the posterior distributionusing the standard MH algorithm. The distribution P. If instead one used a diffusion approximation based method it would be necessary to sample from a posterior distribution of much higher dimension  we observe l measurements that are proportional to the number of RNA molecules either from a single cell or from a population of s cells. This gives a (n + 1) × l matrix of data pointsIn the case of reporter assays based on Polymerase Chain Reaction  measurements are obtained from the extraction of the molecular contents from the inside of cells. Since the sample is sacrificed, the sequence of measurements are not strictly associated with a stochastic process describing the same evolving unit. Assume that at each time point λ is the proportionality constant, ti. j = 1,..., l indexes the l measurements that are taken at time ti. Note that t denote the distribution of measured RNA at time t (ut ~ ϒt). In order to accommodate for the different form of data we modify the estimation procedure as follows. For analytical convenience we assumed that the initial distribution is normal μt and where ψ data set using SSA algorithm with parameter values given in Table b4/γR. The estimation results in Table where kP. The model with the LNA isIn this section the method is applied to experimental data. After a period of transcriptional induction, translation of gfp was blocked by the addition of cycloheximide (CHX). Details of the experiment are presented in Additional file λ. For comparison we also consider the diffusion approximation for which an exact transition density can be derived analytically (see Additional file The observed fluorescence is assumed to be proportional to the signal with proportionality constant Since incorporation of measurement error for the diffusion approximation based model is not straightforward, we assume that measurements were taken without any error to ensure fair comparison between the two approaches. Table The aim of this paper is to suggest the LNA as a useful and novel approach to the inference of biochemical kinetics parameters. Its major advantage is that an explicit formula for the likelihood can be derived even for systems with unobserved variables and data with additional measurement error. In contrast to the more established diffusion approximation based methods ,20 the cThe suggested procedure here is implemented in a Bayesian framework using MCMC simulation to generate posterior distributions. The LNA has previously been studied in the context of approximating Poisson birth and death processes -24,35 anMK proposed and implemented the algorithm. CVH performed the cycloheximide experiment. MK wrote the paper with assistance from BF and DAR, who supervised the study.Supplemental information. Supplementary information contains derivation of the theoretical results, details about algorithm implementation and comparison with the inference method based on the diffusion approximation.Click here for file