Boundary vector cells in entorhinal cortex fire when a rat is in locations at a specific distance from walls of an environment. This firing may originate from memory of the barrier location combined with path integration, or the firing may depend upon the apparent visual input image stream. The modeling work presented here investigates the role of optic flow, the apparent change of patterns of light on the retina, as input for boundary vector cell firing. Analytical spherical flow is used by a template model to segment walls from the ground, to estimate self-motion and the distance and allocentric direction of walls, and to detect drop-offs. Distance estimates of walls in an empty circular or rectangular box have a mean error of less than or equal to two centimeters. Integrating these estimates into a visually driven boundary vector cell model leads to the firing patterns characteristic for boundary vector cells. This suggests that optic flow can influence the firing of boundary vector cells. Over the past few decades a variety of cells in hippocampal structures have been analyzed and their function has been identified. Head direction cells indicate the world-centered direction of the animals head like a compass. Place cells fire in locations associated with visual, auditory, or olfactory cues. Grid cells fill open space like a carpet with their mosaic of firing. Boundary vector cells fire, if a boundary that cannot be passed by the animal appears at a certain distance and world-centered direction. All these cells are players in the navigation game; however, their interaction and linkage to sensory systems like vision and memory is not fully understood. Our model analyzes a potential link between the visual system and boundary vector cells. As part of the visual system, we model optic flow that is available to rats. Optic flow is defined as change of lightness patterns on the retina and contains information about self-motion and environment. This optic flow is used in our model to estimate the distance and direction of boundaries. Our model simulations suggest a link between optic flow and the firing of boundary vector cells. Populations of neurons within the entorhinal cortex and subiculum have firing patterns that depend upon the distance and angle of boundaries in the environment, such as barrier walls. Neurons with this pattern of firing are referred to as boundary vector cells (BVCs) The firing fields of BVCs at a distance from the walls  cannot be explained by proximal cues such as those provided by the whisker system. At least three alternative cues could provide the information for distant firing, and these cues are not mutually exclusive. The first possibility is that distance estimates could be retrieved from memory in combination with sensorimotor path integration. This would require the memorization of the entire environment, especially its boundaries. Information about the spatial location of a boundary would be combined with the current spatial position and head direction to estimate distance and direction of that boundary. The current position and head direction of the rat would be estimated from temporally integrated sensorimotor signals. The second possibility is that multiple visual cues on the wall could be used by rats to estimate the normal of the surface and its distance based on the feature's relative size on the projection, requiring knowledge of the absolute size of the feature. However, typical rat experiments lack the presence of distinct visual features, e.g. wallpapers that could be used to estimate the distance of the wall. Therefore, this possibility seems unlikely. A third possibility is the use of optic flow, the varying patterns of light on the retina while the rat is moving. Optic flow could be used for distance and direction estimation of walls based on the following two assumptions: (i) walls are orthogonal to the ground; and (ii) these walls have piecewise smooth surfaces. In this article, we test this “flow-influence” hypothesis by simulating a rat's trajectory in a circular or square box while estimating the distance and direction of walls from optic flow and integrating these estimates into a model of BVC firing. A priori it is unclear if distance and direction estimates extracted from optic flow are accurate enough to support the firing of BVCs. We demonstrate that these estimates are sufficiently accurate, even for drop offs that lack an orthogonal wall.Further evidence for our flow-influence hypothesis is provided by the rat brain structures processing visual image motion. For instance, neurons in primary visual cortex are sensitive to visual motion A sketch of our model is shown in We make several assumptions to focus our modeling effort on the estimation of self-motion, distance and direction from optic flow. First, the visual field, across a full range of angles extending 240° horizontally and 120° vertically, is simulated using a spherical camera model that describes the flow of individual features of the visual scene by temporal changes of the azimuth and the elevation angle of these features see . Second,Several aspects distinguish our model from previously published template models. In our model, self-motion is restricted to curvilinear motions: These are translational motions along the optical axis combined with rotations around the vertical axis (yaw-rotations). Our template model uses a spherical camera model that helps to account for effects in large visual fields in contrast to a pinhole camera that is restricted to a 180° visual field. Another difference from existing template models is the introduction of templates that are tuned to the combination of self-motion and smooth surfaces modeling walls or ground. This extended tuning allows not only for the estimation of self-motion but also for the estimation of the distance of these surfaces. We make no assumption with respect to the shape of the box, e.g. it could be square, rectangular, or circular. Note that the introduction of multiple models for ground and wall surfaces also requires the segmentation of flow into these separate surfaces. For instance, a rectangular box consists of a ground plane surrounded by planar walls whereas each individual optic flow sample has to be identified as either originating from ground or wall. Given analytical flow for a spherical camera model, the flow that is induced by linear or rotational motion of a wall can be distinguished from flow that is induced by the same motion of the ground. Thus, segmentation in our model is achieved by deciding whether the wall or ground flow template fits better to the sensed flow vector. Our model provides several extensions to existing template models and is motivated by the need to test whether physiological findings of boundary vector cell firing can be explained by using optic flow as a distal cue.We organize our results into four sections. First, we start with the generation of simulated rat trajectories based on the velocity statistics of recorded rat trajectories. These simulated trajectories are used to generate the analytical spherical flow representing the flow that would occur on the rat retina: This flow is then provided as input to our template model. In the second section, we show examples of the wall-ground segmentation and detection of drop-offs with our template model. Third, examples for the estimation of distance and direction of walls are shown, together with the error statistics of distance estimates for an entire simulated rat trajectory. In the fourth section we link our template model to the BVC model and show the resulting firing of model cells compared against data Our aim is to simulate the rat's body movement in an environment similar to the one used in the study of Lever et al. For the generation of rat trajectories we combined a deterministic algorithm with a random component. We randomly generate a linear or rotational velocity that follows a Rayleigh or normal distribution, respectively. As the deterministic component we calculate a rotation that turns the rat to continue to walk parallel to the wall. This turn happens only if the rat is closer than 2 cm to the wall and its head direction is smaller than 90° with respect to the normal vector of the wall. Before the distance of walls can be estimated, flow samples of walls have to be segmented from flow samples of the ground. This is accomplished in the first stage of our model, see In case of drop-offs our model employs a different mechanism by detecting the flow transition from large to small magnitude. Examples of this detection are shown in nd and 3rd column of Instead of modeling specific surface types, like curved and planar, we approximate arbitrary surfaces locally by planes. This allows us to use the same model for curved walls of a circular box or planar walls of a square box. Our template model allows for the estimation of distance and direction to multiple walls. In In addition to these examples of single distance estimates, we evaluated the distance error systematically for each sample point of our simulated rat trajectories of approximately 20 min duration that include 60,000 sample points for the 50 Hz sampling frequency. We compute the mean distance error for each location, computed for all distance estimates made at that location. Note, that this error measure is, mostly, independent of the actual distance to the wall since all positions provide at least two different distances to walls, excluding the center in the circular box or square box. Here, we assume larger distance estimates for the center and smaller ones for areas close to the wall as our model tends to estimate distance to closer walls rather than farther walls. For the circular box, the mean error is largest in its center; see Optic flow could be one cue to support firing of boundary vector cells (BVCs) that fire for walls being present at a specific distance and allocentric direction. Distant firing distinguishes BVCs from border cells We compare the data of recordings of BVCs any wall of an allocentric direction and distance. In our simulation, firing appears also next to the inserted wall. In the same way our model cells would adapt to wall changes in the environment as shown in another experiment which involved nesting two boxes, a small one in a bigger one. After some time the smaller box is quickly removed in that experiment. Then firing of BVC shifts its absolute position in the larger box to resemble the same distant tuning that it had in the small box Aside from using a square and circular box we provide additional simulations with a wall inserted inside of the square box and in another simulation we removed all walls to model a platform. For a platform, the drop-off is detected as a local discontinuity in flow direction and speed. Once the elevation of the drop-off is determined, it is converted into a distance estimate. All distance estimates are fed into the BVC model with their response values. This paper presents a template model for scene-segmentation and the estimation of geometric properties of the environment, namely the distance and allocentric direction of walls and drop-offs. Distance estimates of our model in empty boxes are accurate within a two-centimeter-range; for a square box with an inserted intrinsic wall the error is higher at locations at the inner edge of the narrow passages created at either end of the intrinsic wall. When these distance and direction estimates are integrated into the boundary vector cell model Template models for the estimation of ego-motion have been used mainly as a model for self-motion estimation in primates Most studies on firing properties of hippocampal structures in rats focus on visual cues in general, e.g. a cue card, but not on optic flow as such. For instance, visual cues influence the orientation and firing location of hippocampal place cells Although visual input provides a rich set of information, other cues might be important for BVC firing, as well. Vestibular information and visual motion influences hippocampal place cell firing Three alternative technical solutions are possible for the segmentation of walls from ground, and subsequently the estimation of self-motion and environment variables. These are: RANSAC For real-life images the quality of flow based segmentation depends largely on the quality of the detected flow and the dissimilarity between flow templates or flow vectors at the drop-off. Since we do not know the quality of detected flow for real-life images we study simulated noise superimposed on the analytically defined flow. Examples with additive Gaussian noise in each component of the flow with a signal-to-noise ratio of approximately 70 dB leads to larger errors in distance and direction estimates see . A majorFurther properties of our model are the logarithm used in the matching function and the model's capability to incorporate tilt angles. Choosing a logarithmic sampling and the logarithm of motion speeds to compare input flow vectors and template flow vectors makes sense for a first-person perspective from an ecological and behavioral point of view. Typically, objects' distances that are far do not have to be represented with a high sampling, e.g. of centimeter-precision, because they are not reachable or are not potential obstacles. A logarithmic sampling of distance values also has an effect on the comparison between flow vectors of different distance. For optic flow generated by translational self-motion the length of flow vectors is inversely proportional to the distance of a sample point in 3D space. By transforming these distances using a logarithm we put more emphasis on short flow vectors that relate to points that are close to the rat. Another property of our model is the incorporation of non-zero tilt angles. In such configurations the optical axis is not parallel to the ground. The normal vector that describes the wall or ground becomes dependent on the head direction. In our model this head direction is assumed to be given, e.g. by the vestibular cues captured by the head direction cell system, as is the tilt angle see also . Then ouOur current model has several limitations. So far, our model responds only to visible walls and drop-offs; however, place cells that may be driven by BVCs also respond in the presence of transparent walls Information about self-motion and environment structure that is extracted by our model from optic flow could be useful for other cell types as well. Grid cells can be generated by temporally integrated linear and rotational velocities that are estimated from optic flow Following our “flow-influence” hypothesis our model would predict cells with sensitivity to large flow fields. However, instead of in hippocampal or related areas, we assume this sensitivity to exist in sensory related areas, such as the primary visual area or higher level visual cortical areas or the accessory optic system. In primates these sensitivities have been found in area MT and MSTd We divide the explanation of our template model into the following steps: First, we define the spherical image flow model for curvilinear self-motion. In the second step, flow templates are defined for a ground-plane and planar walls. Third, tuning or matching functions for the comparison between input flow and template flow are defined. Then we summarize all computational steps in an algorithm. Fourth, to interpret and visualize the representation of our template model we define a read-out method. Fifth, a description for the integration of estimated distance and direction values into the boundary vector cell model is given. Parameters of the spherical camera model, template model, and boundary vector cell model are summarized in The spherical image flow model for instantaneous motion through a rigid stationary environment is In the simulations we assume that the rat is moving tangent to the recorded trajectory in the 2D plane. This assumption reduces the six degrees of freedom of the model to two degrees of freedom: The linear velocity γ = 0 and distance γ≠0 the normal vector is given by ϕ. This normal vector In Hessian normal form a plane is described by its unit normal vector ϕ of the rat's head which results in α being the allocentric direction of the wall. For a tilt angle γ≠0 the wall's normal vector is described byα, second for the tilt angle γ, and third by the allocentric direction of the rat's head ϕ.The depth function for planar walls assumes a wall to be defined by the normal The distance function from Equation (3) with the corresponding normal vectors is plugged into Equation (2) to define the template flows for curvilinear self-motions defined by yω but depends on the distances D, we multiply the Equation (5) by the vector Input flow is defined as m velocity samples. For the wall-ground segmentation we use the following decisions to define the set of wall samples Second, the tuning function for potential sample points of walls is defined by:Then we continue with the ground samples to estimate the linear velocity of the rat by using the tuning function:Next, we compute the rotational velocity from ground samples. For this computation we use the following tuning function which computes the Euclidean distance between input flow and template flow:In the last step we estimate wall distances for a given allocentric direction and use the already estimated linear and rotational velocity from Equation (8) and (9). For this distance and direction estimation we use the tuning function as defined by Perrone Our extended model can be summarized into the following four steps. First, we compute a wall-ground segmentation by using the tuning functions from Equation (6) and (7). The segmentation is determined by whether a flow vector fits better to a ground template vector from Equation (6) or a wall template vector from Equation (7) while sampling all possible linear velocities in Equation (6) and all possible allocentric directions and possible distances for a wall in Equation (7). Therefore, the segmentation is computed without knowing the parameters of self-motion. But once the segmentation into wall-ground is known we use ground samples to estimate linear and rotational velocity in step two and three, respectively. For estimating linear velocity we use the tuning function from Equation (8) and for rotational velocity the tuning function from Equation (9). In the fourth step, we estimate distance and allocentric direction of walls using the known segmentation, linear, and rotational velocity. A pseudo-code of the algorithm is provided in the So far, we have not described how velocities, distances, and directions are estimated given the activity from evaluating the residual functions in Equation (8), (9), and (10). Such a description follows.The matching functions in Equation (8), (9), and (10) depend on different stimulus parameters. For instance, the function of Equation (8) depends on linear velocity samples, whereas the function of Equation (10) depends on distance and direction of walls. Our read-out distinguishes between 1D and 2D functions. For a 1D function our read-out method uses a weighted sum with two percent of all argument values that are centered on the maximum. For the 2D match function of Equation (10) we use a different method. Our read-out mechanism selects all matches with their value being within the 70% range with respect to the maximum match. These match values together with their respective arguments, in the above example the linear velocities, are passed along to the distance error calculation or BVC model. The calculation of distance errors takes the direction arguments and computes the ground-truth distance for this direction. Then this ground-truth distance is subtracted from the estimate. The absolute value is computed for this difference to define the distance error.If the boundary vector cell model is the next stage, arguments about distance and max read-out directions are passed together with their activation. Distance and direction are integrated into the existing BVC model and we weigh each BVC activity by the match value provided by our template model. This is described in more detail next. A pseudo-code of the 1D and 2D readout method is given in i-th BVC is tuned to the distance k and j range over all read out activations from our model that are above 70% of the maximum activity. Parameter values for this Equation (11) and all other equations are reported in The boundary vector cell (BVC) model was described in detail elsewhere Drop-offs are detected with a center-surround filter applied to the speeds of the flow field. This detection method assumes that the azimuth and elevation angles are arranged on a regular sample grid that is associated with pixels. In our model example we use 400 horizontal samples and 200 vertical samples. To detect the flow discontinuity at the drop-off we apply a center-surround filter kernel to the length of the flow vectors and detect its maximum response. Formally, this is expressed by:h, tilt angle γ, and head direction ϕ are known:The detected elevation angle of the drop-off is then converted into a distance estimate assuming the eye-height Figure S1θN and ϕN drawn from a normal distribution with zero mean and nσ as standard deviation. This type of noise strongly influences the quality of the segmentation and, thus, influences the other estimations based on this segmentation. In both examples the rat is positioned at 0x = 20 cm, 0y = 3.5 cm, 0z = 15 cm, has the head direction ϕ = 23°, and the self-motion zv = 19 cm/sec and yω = 25°/sec. a) Square box with the coordinate system. The distance estimates are indicted by magenta arrows. b) Spherical flow field with noise (nσ = 5°/sec) and wall-ground segmentation. c) 2D match values as calculated from samples that are indicated as originating from a wall, these are the blue dots in b). Low match values are encoded by low intensity and high match values by a high intensity, see also the inset with the color code. d–g) 1D match values for the linear and rotational self-motion and the direction and distance of walls. The latter two curves are computed as the maximum response from the 2D match value function of c), whereas the maximum is computed for the dimension not shown. The estimated self-motion is z,estv = 19 cm/sec and y,estω = 24°/sec and the mean distance error is 1.15 cm with a standard deviation of 0.52 cm. h) Circular box with coordinate system and distance estimates of the wall depicted by the magenta arrows. i) Spherical flow with noise (nσ = 15°/sec) and wall-ground segmentation for this circular box. j) 2D match values. k–n) Match value functions for velocities of the camera and direction and distance of walls. The estimated self-motion is z,estv = 19 cm/sec and y,estω = 25°/sec and the mean distance error is 3.03 cm with a standard deviation of 1.29 cm. The distance error in the circular cage is higher due to the assumption about a planar approximation for each segment of the curved wall.Examples of wall-ground segmentations, distance, and direction estimation of walls for analytical flow superimposed with additive, independent Gaussian noise. Flow with noise is defined as (TIF)Click here for additional data file.Figure S2a) Shows the spherical flow field and the detected wall-ground segmentation. b) 2D match values for the matching using the log-speed difference shows three clearly distinctive high intensity regions. These are better visible in the break-down into 1D curves c) for the wall's direction taking the maximum of all distances and d) the wall's distance taking the maximum of all directions. The mean error in distance estimates is 2.76 cm with a standard deviation of 1.81 cm. e) 2D match values for the matching using the speed difference without applying the logarithm. Compared to b) the matching occurs fuzzy, also visible in the break-down in the matching for f) the walls' directions and g) the walls' distances. In this case the mean distance error is 3.74 cm with a standard deviation of 3.85 cm and, thus, higher than in b). This example uses a rectangular box 250 cm×280 cm with 50 cm high walls and the rat's position is 0x = 0 cm, 0y = 3.5 cm, 0z = 10 cm with the head direction ϕ = 15°.Taking the logarithm of the speeds of input and template flow vectors as the argument for a Gaussian function provides a “sharper” tuning compared to taking only the difference of speeds without the logarithm. (TIF)Click here for additional data file.Figure S3a) Simulated BVC firing for a square box with distance and direction estimates for walls provided from our template model. b) Mean distance error in the two-centimeter range. c) Simulated BVC firing for the same square box as in a) but with an additional intrinsic wall. Firing appears at any wall of a specific distance and allocentric direction the cell is tuned for. d) Mean distance error in the range of zero to six centimeters. e) Simulated BVC firing for a circular box. f) Mean distance error in the two-centimeter range.A tilt angle of 30°, that is the angle between the optical axes compared to the horizontal parallel to the ground, does no change firing fields qualitatively; however, the error at larger distances is larger compared to a zero tilt angle. (TIF)Click here for additional data file.


Direct sunlight absorption by trace gases can be used to quantify them and investigate atmospheric chemistry. In such experiments, the main optical apparatus is often a grating or a Fourier transform spectrometer. A solar tracker based on motorized rotating mirrors is commonly used to direct the light along the spectrometer axis, correcting for the apparent rotation of the Sun. Calculating the Sun azimuth and altitude for a given time and location can be achieved with high accuracy but different sources of angular offsets appear in practice when positioning the mirrors. A feedback on the motors, using a light position sensor close to the spectrometer, is almost always needed. This paper aims to gather the main geometrical formulas necessary for the use of a widely used kind of solar tracker, based on two 45° mirrors in altazimuthal set-up with a light sensor on the spectrometer, and to illustrate them with a tracker developed by our group for atmospheric research. In the UV-visible range, light scattering is more important and enables spectroscopic studies of the atmosphere in other geometries such as zenith measurements )2 atmospheric loading, in the framework of the new Total Carbon Column Observing Network (TCCON). The geometry of the Sun tracker is altazimuthal. It was built at our institute and used to validate the methods described in the last section.A second FTIR station has been installed in Saint-Denis in September 2012. This station, which is also automated, is based on a Bruker 125 HR spectrometer, more appropriate to measure COMtracker on its rigth side by the corresponding factors derived from This new solar tracker uses a Newport RV-160 rotation stage for the azimuth rotation and a Vexta stepping motor with a gear box for the altitude. Both rotations are driven by a Newport XPS controller, linked to the controlling PC. The tracker mirrors are elliptical with a 10 cm minor axis. The photodiode setup was purchased from Bruker with the spectrometer and is installed at the input window of the spectrometer. It consists of a 1 cm mirror which reflects a small portion of the incoming light to a 18 cm focal length lens which focuses the beam onto the 4-quadrant photodiode. The optical path is shown in α,β and γ described in Section 2.2 were respectively estimated to be 9.96°, 0.888° and 0.174°. The right panel shows how these fitted angles improve the calculated Sun position. The maximum offset between calculated and actual position is now 0.5°, i.e., 9 mrad, which is under the 20 mrad of the photodiode's FOV. Providing this accuracy in the calculated mode, the tracking system is able to set the Sun's image onto the 4-quadrant photodiode and then start the active tracking without the need of an operator.5.We have derived the geometrical formulas needed to track the Sun with a kind of altazimuthal tracker widely used in atmospheric remote sensing. The setup is based on two rotating 45° mirrors facing each other and a 4-quadrant photodiode involved in a closed-loop control of the tracker. After discussing the required accuracy for the calculated mode and calculating the FOV of the sensor, we described how to take into account and estimate the Euler angles, representing the orientation of the tracker compared to the ground. These sections can actually be applied to other tracking setups. On the other hand, even if the method is general, the formula for the active tracking depends strongly on the optical configuration and may not be used for other trackers' geometries. We have proposed a control loop with PID to achieve a smooth tracking while reducing overshoot and the residual part of the error. Finally, we have tested the formulas with a custom-built solar tracker that has been installed together with a FTIR spectrometer at Reunion Island in September 2011.Among the future work will be the improvement of the tracking smoothness, and particularly the tuning of the six parameters of the PID controllers. We will also implement the solution presented in Section 3 and check whether the measurement dead time can be reduced.A characteristic of the Maido observatory is the very regular cloud cycle. At noon, the clouds reach the observatory almost every day. This is very convenient for clouds studies but less for solar occultation trace gases measurements. On the other hand, the nights are so clear up there that the observatory was first supposed to be dedicated to astronomical research. This is thus a good place to try Moon tracking and we plan to work on that in the future.


Protein–protein interactions in cells are widely explored using small–scale experiments. However, the search for protein complexes and their interactions in data from high throughput experiments such as immunoprecipitation is still a challenge. We present "4N", a novel method for detecting protein complexes in such data. Our method is a heuristic algorithm based on Near Neighbor Network (3N) clustering. It is written in R, it is faster than model-based methods, and has only a small number of tuning parameters. We explain the application of our new method to real immunoprecipitation results and two artificial datasets. We show that the method can infer protein complexes from protein immunoprecipitation datasets of different densities and sizes.4N was applied on the immunoprecipitation dataset that was presented by the authors of the original 3N in Cell 145:787–799, 2011. The test with our method shows that it can reproduce the original clustering results with fewer manually adapted parameters and, in addition, gives direct insight into the complex–complex interactions. We also tested 4N on the human "Tip49a/b" dataset. We conclude that 4N can handle the contaminants and can correctly infer complexes from this very dense dataset. Further tests were performed on two artificial datasets of different sizes. We proved that the method predicts the reference complexes in the two artificial datasets with high accuracy, even when the number of samples is reduced.4N has been implemented in R. We provide the sourcecode of 4N and a user-friendly toolbox including two example calculations. Biologists can use this 4N-toolbox even if they have a limited knowledge of R. There are only a few tuning parameters to set, and each of these parameters has a biological interpretation. The run times for medium scale datasets are in the order of minutes on a standard desktop PC. Large datasets can typically be analyzed within a few hours. Protein–protein interactions (PPIs) constitute the core of inner cell communication,2. LargeProtein complexes work as functional units in the interaction network, and the members of a complex do not appear individually. For this reason, proteins of the same complex are predicted to have a similar abundance in different IP–experiments. Consequently, a similarity-based cluster analysis of these datasets leads to clusters that represent the protein complexes. The clustering should be based on both parts of the information, namely, the occurrence of proteins in the samples and their relative abundance values. Complex–complex interactions (CCIs) as a coarser view on the PPI network can be represented by a clustering in which clusters are allowed to share proteins as proof of the interaction.Large–scale IP/MS study by Gavin et al. first described such analysis for proteome wide characterization of protein complexes in yeast. SubsequA medium scale IP dataset for the components of the human Tip49a/b–PPI complex was presented by Sardiu et al. and used to illustrate complex inference. SeveralThe different clustering methods are scattered widely, and all studies concluded that there is no standard method available currently that is suitable for clustering all kinds of IP–based interaction data. In addition, some of those methods, for example biclust, involve many parameters and are very time–consuming as they need many iterations to get to a result.Malovannaya et al. introduced a method called Near Neighbor Network clustering (3N) for predicting core protein complexes in their large–scale IP/MS study on human cells. This alIn this study, we present a new universal complex inference method that is based on the idea of 3N clustering which we call "4N". Most of its parameters can be set automatically; moreover, it contains tools for visualizing complexes and their interactions. Major advantages of 4N over other methods include that it can process very large datasets, that it allows an immediate insight into the data and that it directly shows the effect of the parameters on the cluster result. 4N is simple and much faster than methods based on probabilistic models, such as biclust. The high speed of 4N in combination with the visualisation tools allows the quick interactive search for protein complexes. A software toolbox written in R is available in the software section on the website www.bdagroup.nl. It includes the algorithm itself, documentation, running examples and methods for plotting and evaluation of the cluster result based on given reference complexes.The article starts with the description of the empirical and simulated IP datasets that we used for testing 4N. The 3N and 4N methods are described in detail thereafter. The article closes with detailed results for all datasets and a Discussion and conclusion section.U of 0.125 to "malovIP" and removed all proteins that were not assigned to near neighbor networks. The subset contained, besides many other proteins, all protein subunits of the Integrator[For a direct comparison with 3N, we did an analysis using our methods on the very large dataset from the original 3N publication, which ctegrator, Mediatotegrator,21, and tegrator, the "meThe next dataset that we analyzed is the "human Tip49a/b" from. We obtahttp://www.string-db.org,[Unlike PPI networks from yeast–two–hybrid studies, only a few IP–based datasets are available; information about reference complexes especially is rare. We created another small–scale, and a medium–scale IP dataset from two simulated networks for this reason. The two PPI networks were created with help of the protein interaction database string-DB .Snap25 to search for proteins that are connected to it in string-DB. We extended this network further with indirectly connected proteins, until we obtained a network with 73 nodes and 116 edges  with help of the plugin clusterViz (http://chianti.ucsd.edu/cyto_web/plugins/displayplugininfo.php?name=ClusterViz) and using the MCODE method by[We first searched for complexes in these two networks using the software cytoscape , then high for its direct interactants and lower for the indirectly connected proteins. Thus, antibodies that target different complex subunits are expected to pull out slightly different protein subsets from the complex, which leads to redistribution of measured abundances for each protein.Kim et al. presented the following method to simulate this effect on graphs, where aThis paragraph contains a brief description of the 3N algorithm, details can be found in the Additional fileU,V in the IP-matrix as arccos(U ∗ V/||U|| × ||V||) like explained in[3N defines the near neighbor network for a certain protein as the set of proteins that appear reciprocally or together, and have correlated abundance values with this protein across the different-antibody samples where it has its highest abundance. The correlation of two proteins is defined in terms of the cosine-distance between their rows ained in and it hAlthough the original 3N algorithm performs well on the kind of data described in, it has C) is relatively weak for separating proteins that do not belong together, as two random vectors of only positive values are more likely to have a cosine-distance < 65 than one between 65 and 90. The 3N algorithm was not published as software implementation by the authors, which makes it impossible for others to immediately apply it to their own IP data.To calculate the near neighbor network for one protein, 3N considers only the samples where this protein has its highest abundance values . This can lead to a loss of proteins when the length of the Top list (parameter L) is too short. A set of close proteins is only detected as core complexes when all their NNNs contain the complete protein set. Less closely connected complexes easily become predicted incompletely. A threshold of 65 for the cosine-distance . In our analysis it is set to 40 such that the algorithm is able to keep proteins in one cluster that build an unbranched chain in the network. The value was determined experimentally using the two artificial datasets. This parameter has an influence on the absolute number of proteins in NNNs. The other external parameter (P), which is used to decide which core complexes should be joined finally, does not influence the total number of proteins that are put into complexes. Below we give an overview of 4N; a detailed description can be found in the Additional fileOnly two external parameters remain that need to be optimized manually for different datasets. One of them is the cosine-distance threshold parameter  is set to 0.01 initially to assign all proteins that fulfil this requirement to a near neighbor network. After this, U is set to the highest possible value, where each of the proteins still remains in at least one NNN. The cosine-distance threshold (C) is set to 40. The core complexes are calculated from the NNNs with the highest possible value for the Parameter S (Jaccard coefficient threshold for building CCs), for which all proteins occur in at least one core complex. Result of this step is set of core complex proteins for each protein in the dataset. These sets of proteins overlap highly, especially when they are built with a low S. For this reason, we added a step to join the core complexes (step "joinCC") based on the relative number of shared proteins.Joining core complexes We define the overlap between two complexes as the relative number of proteins in the smaller complex that occurs in the larger one. When the threshold for joining core complexes (P) is smallest, all core complexes that share at least one protein are joined, which leads to completely distinct sets of joined core complexes. They represent groups of core complexes that interact within but not across the groups. Higher thresholds create smaller sets of partly overlapping complexes that stand for core complex isoforms, a value of 1 creates sets that represent MEMOs. Here, two complexes are only joined when the larger one contains all proteins of the smaller one.P in a range from 0 to 1 are used to create a core complex plot alike to the one in FigureP, and the matrix cells for proteins that are in the same joined core complex become 1, shown by the large non-overlapping squares in FigureP and the matrix cells for proteins that are still in the same core complex are incremented in each step. A standard hierarchical clustering with Euclidean distance is applied to each of the large squares to display a heatmap of the inner structure and the overlap of the joined core complexes. The image shows the MEMOs as well as the core complex isoforms and can be used for visualisation of the complex assignments for each protein. It also shows whether the automatic value for U is high enough to distinguish between the complexes. A too low U leads to one very large square in the plot with large white areas, see the plots for "Tip49a/b" in the Additional fileU, the plot can be used to set P to decide whether the results should contain larger, less overlapping complexes or smaller dense complexes that share proteins. The output of 4N for one specific set of parameters U,C,S,P is a list of predicted complexes. Each list element contains the names of the proteins that are in the complex.Different values for The reference complexes for the three real datasets  are those mentioned in the corresponding paragraphs in the Data description section. The reference clusters for the simulated data  are those that were found by MCODE in the equivalent networks.The threshold for joining the core complexes was selected for each dataset by examining the corresponding core complex plots and set to 0.5 for the artificial datasets, 0.6 for "Tip49" and to 0.85 for the tests on the "malovIP" datasets. The complex prediction quality was tested by comparing the 4N results with the reference clusters using the method by Brohée and van Helden.G be the set of given clusters, P the set of predicted ones. Each given cluster g ∈ G is seen as a set of proteins, likewise each predicted cluster p ∈ P. The scoring method uses the Jaccard coefficients for each combination of p and g as |p ∩ g|/|p ∪ g| and sets them in relation to the sizes and number of the real and predicted complexes. A complete description of the method can be found in[Both the given and the predicted clusters can be seen as sets of sets. Let found in. The metA perfect complex prediction yields 1 for the accuracy score which means, that all reference complexes are completely covered by predicted complexes and the predicted complexes only contain proteins of their corresponding reference complex. The separation score is 1 when each reference complex is covered by exactly one predicted complex.P) was set to 0.85 after examining the core complex plot. The predicted complexes were compared to the 10 reference complexes. On both datasets, the same scores were reached as all relevant proteins are in the "malovIP_subset" and the additional number of proteins in the "malovIP" dataset did not influence the prediction. FiguresThe two real datasets "malovIP" and "malovIP_subset" were analyzed and the joining-parameter , it is very fast and can process very large IP datasets. The 4N analysis on the "malovIP"-dataset with its approx. 3200 IPs and 11500 proteins took approximately one day , the analysis of "Tip49a/b" can be performed in seconds.C) remains instead of four parameters as in the original 3N algorithm. The immediate feedback given by the core complex plot allows assessment of the result and a manual adaptation of the parameters if necessary. 4N can process datasets of different density from dense to sparse. It is easy to install and execute as only a basic R-installation and one extra library is needed as minimum requirement. The method can efficiently cluster IP data and suggest protein complex compositions.4N predicts the reference complexes in all tested datasets with high accuracy. The algorithm allows an immediate insight into the general density of an IP-dataset, the MEMOs, the core complex isoforms and the complex-complex interactions with the core complex plot. When this plot is seen as result, only one parameter that influences the result (the cosine-distance, The authors declare that they have no competing interests.JK implemented the software and performed the calculations. In the joint writing of the paper IvM focused on the mathematical background, AM and ABS concentrated on the biology and HH and AKS coordinated the whole project. All authors read and approved the final manuscript.Supplementary material.Click here for file


Biological protein interactions networks such as signal transduction or gene transcription networks are often treated as modular, allowing motifs to be analyzed in isolation from the rest of the network. Modularity is also a key assumption in synthetic biology, where it is similarly expected that when network motifs are combined together, they do not lose their essential characteristics. However, the interactions that a network module has with downstream elements change the dynamical equations describing the upstream module and thus may change the dynamic and static properties of the upstream circuit even without explicit feedback. In this work we analyze the behavior of a ubiquitous motif in gene transcription and signal transduction circuits: the switch. We show that adding an additional downstream component to the simple genetic toggle switch changes its dynamical properties by changing the underlying potential energy landscape, and skewing it in favor of the unloaded side, and in some situations adding loads to the genetic switch can also abrogate bistable behavior. We find that an additional positive feedback motif found in naturally occurring toggle switches could tune the potential energy landscape in a desirable manner. We also analyze autocatalytic signal transduction switches and show that a ubiquitous positive feedback switch can lose its switch-like properties when connected to a downstream load. Our analysis underscores the necessity of incorporating the effects of downstream components when understanding the physics of biochemical network motifs, and raises the question as to how these effects are managed in real biological systems. This analysis is particularly important when scaling synthetic networks to more complex organisms. Cells rely on complex networks of protein-protein interactions in order to carry out life functions. Scientists believe that these networks are organized in a modular fashion; that is they are made up of functionally distinct parts like an electronic circuit. Modularity implies that just as we put together electronic parts to make an amplifier that we can use in many different circuits, we can put together biochemical reactions to make an amplifier, or a switch or an oscillator, which perform the same function in different organisms. This assumption is important in synthetic biology, where we engineer and assemble synthetic genetic circuits in living organisms in a modular fashion. We show that for important modules like genetic and signaling switches, the assumption of modularity has a crucial limitation. We show that if one simply connects a biological switch to another downstream circuit, the presence of the connection changes the operation of the switch, which in some cases may stop behaving like a switch. Our work underscores the importance of taking into account these downstream connections and suggests that natural systems may be balancing some of these components in order to ensure that despite diversity, modules continue to work in different systems with fidelity. A longstanding question about signal transduction and gene transcription networks is how modular are they. Here modularity means relative insulation of small subgraphs or motifs of the main network from each other For synthetic biologists the challenge is now to move from simple network motifs such as pulse generators Loads could also have noteworthy unrecognized effects in natural systems. In fact all natural systems have loads in some ways or the other. Motifs in signal transduction networks are connected directly to a transcriptional response, or to downstream proteins that may function as transcription factors or go on to activate transcription factors. Motifs in gene transcription networks have transcriptional outputs with protein domains that bind nonspecifically and specifically to binding sites on the DNA, apart from interacting with other transcription factors.Circuits that function as switches play an important role in all biological signaling and gene transcription networks because they encode decisions. This change of state can be brought about by an external signal, or an internal accumulation of a protein, which can drive the system to a different steady state. Examples are the regulatory circuits for the cell cycle in yeast B. subtilis, ComK represses the production of Rok and Rok represses the production of ComK; however ComK also has a strong positive feedback upon its own production These reasons prompted our theoretical study of the behavior of a simple genetic toggle switch The Ras protein is a G-protein found on mammalian cellular membranes that is important in many cellular processes and is an upstream activator of the MAPK pathway. Ras goes from a GDP-bound inactive form to a GTP-bound active form, often in a digital manner 1T is a constant, L1 is the free load and conservation gives us the bound load as L1T−L1.The basic genetic toggle switch consists of two mutually repressing genes as shown in 1 = α2 = 0.2; β1′ = β2′ = 4; n = 3; kon1′ = kon2′ = 0.5; koff1′ = koff2′ = 0.5; k1 = k2 = 1; [L1T] and [L2T] are variable. Note that These four equations are presented in de-dimensionalized form, with To calculate transition times, we first start the system in one state, say high Repressor 1. After the system has reached steady-state, we add a constant concentration of the inducer and measure the time taken for Repressor 2 to go from 10% of its maximum value to 90% of its maximum value. This is the “rise time”. Similarly the “decay time” is the time taken for Repressor 1 to go from 90% of its maximum value to 10% of its maximum value. The level of the inducer remains fixed.In practice the inducer may decay and the transition would depend upon there being inducer present for a sufficiently long time to induce transition. In such cases the amount of inducer required may be of interest. When the inducer is applied as a bolus with a first order decay rate, it appears as an exponentially decaying pulse. We thus included a fifth differential equation governing the amount of Inducer.Here A genetic toggle switch can be induced to change states by the alternative method of repressing the highly expressed repressor, and in fact the original toggle switch used this form of induction Here Our model assumes that when the repressor protein is bound to the load, it is protected from degradation. However it is also possible that even when the protein is bound to the load, it can still degrade. To check the impact of removing the protection assumption, we also consider an additional model where the repressor can still degrade with the same rate constant when bound to the load. The equations for that model are slightly modified versions of the equation above, and are presented in detail in the Supplementary We conducted parameter sensitivity analysis on models utilizing both forms of induction; these did not show any qualitative change on wide variation of key parameters .A positive feedback was added to the R1 side of the toggle switch as an inducible promoter with a Hill coefficient of 1. We assumed that the positive feedback acted on the same promoter as the repression, resulting in a composite term for production of R1 from promoter 1 where ρ is the strength of positive feedback.1 is the leaky production of R1 while α1+β1 represents the activity of the promoter in the absence of repression or positive feedback. We chose k2 and k5 = 1, d1 = 0.2, and for the figures in the main paper we chose ρ = 3.5. We address other values of the positive feedback in The derivation of this equation can be found in the Supplementary We perform stochastic simulations and histogram the concentrations of the repressor proteins to construct their probability distribution. The quasi-potential of the toggle is given by the negative logarithm of this probability distribution Stochastic simulations were carried out using a modified Gillespie algorithm using the standard rate expressions for every reaction . We chosFor the positive feedback toggle switch the same equations were used except for the repressible production of Repressor 1, where we used instead the rate expression given by the right hand side of For our study we adapted the minimal model of the Ras switch proposed by Das et. al. After Raf binds RasGTP, the complex catalyzes the phosphorylation of Raf leading to a phosphorylation cascade. For this study we ignore Raf activation and only consider the effects of Raf as a binding partner for RasGTP. The Das paper We use the following notations for the species involved in the system:Moreover, four of the basic protein species along with the complexes they participate in have associated conservation laws. These are as follows:In the Ras model too we implicitly assume that when RasGTP is bound to Raf, it is protected from de-activation by a RasGAP. We also study the effects of relaxing this assumption on both the LMA and the PSSA models. The modifications to the original model are detailed in the Supplementary We used XPPaut to perform a bifurcation analysis of the Ras switch with changing levels of SOS, with and without a load. The quasi-potential landscape does not provide useful insights into load induced modulation of the Ras switch and hence the probability distributions are not reported.The presence of a binding partner for either Repressor 1 or Repressor 2 (which we refer to thereafter as the load) introduces new terms in the differential equations describing the toggle switch, i.e. the last two terms in The exception is when the repressor molecule can degrade even when bound to the load, which may be relevant in some experimental situations. As The reason for the change in steady state behavior is made clear on examining the equations of the system. Here we need to incorporate additional reactions that represent the decay of the repressor-load complex into the load alone. This leads to an additional term in the equation for the load and the repressor-load complex (Eq. S44 and S45). However this term does not appear in the equation for the repressors, which continue to be governed by As can be seen from an examination of the chemical reaction system, this mechanism of abrogation of bistability arises whenever the load-repressor complex participates in a non-reversible (from the repressor's point of view) chemical process that leads to an unbalanced leakage of the repressor from its function as a repressor by the presence of the load. A more interesting example of such a process could be provided by a chemical reaction system where the load is an enzyme for one of the repressor molecules, which is transformed by the enzymatic action into a protein no longer capable of repression. The mathematical analysis of this case is exactly the same as the model we are currently discussing hence we do not consider it separately here.However a load can significantly change the dynamic response of the basic genetic toggle switch as we shall see below. We examined two different measures of dynamic response, response time for state switching and the amount of inducer required for state switching.We measured two response times, the rise time which quantifies the time taken for the concentration of Repressor 2 to increase from its low or zero level in state 1 to its high level in state 2, and the decay time which measures the time taken for Repressor 1 to decay from its high level in state 1 to its low level in state 2, in both cases in response to a constant inducer. Specifically the rise time measured the time to go from 10% to 90% of the steady state maximum, while the decay time measured the time to go from 90% to 10% of the steady state maximum. These measurements were made using the deterministic model in the cases when the load was applied only to one side and to both sides of the switch.We found that both the rise time and the decay time increase with increasing load concentration. Interestingly, this relationship was approximately linear in all cases . The sloWhile we also found an approximately linear relationship between the decay time and the concentration of the load, there was little difference between the decay times for the state with the load (“same side load”) and the state without a load (“opposite side load”) at our base parameter values. Thus the load affects rise time and decay time differently. When a load was applied to both sides of the switch, the slope of the decay time linear fit was larger, again indicating the operation of both delays.We tested these results by changing parameter values for the binding of the load  and founPrevious studies of response times of biochemical networks with and without a load have also seen monotonic increases in the response time of simple transcriptional circuits An increase in response time should also imply that the concentration of inducer required to shift states should also be affected, especially when it can decay. In accordance with this expectation we also found that the concentration of inducer required to switch states increased exponentially with increasing load, as seen in In our analysis so far we have assumed that the total concentration of the load is fixed. We now analyze the case when the load is generated by a constitutively active promoter and can decay at a first order rate. We find that in this case too the qualitative features of the transition time remain the same as the toggle switch with a fixed load, i.e. it was approximately linear in all parameter regimes tested .This can be seen in another way in Thus the addition of the Raf scaffold, which is an integral part of the MAPK cascade, fundamentally changes the qualitative behavior of the positive feedback switch. The main reason why the steady state bifurcation properties are affected here in contrast to the basic genetic toggle switch is that for this signaling circuit, as seen in To see this more generally, consider for example a chemical reaction system comprising of n-species Note that for simplicity of notation we have not indicated the dependence of the dynamical system on its own parameter values. Now in the steady state, if the set of equations is complete, the left side uniformly goes to zero and we recover the result that the steady state remains exactly the same with or without a load, as for the genetic toggle switch. However let us now assume that we have an additional conservation law, say,This conservation law implies that one equation in our dynamical system is redundant, and we need to drop one equation to make the system linearly independent. We can decide to drop It has been pointed out previously that significant sequestration effects can abrogate zero order ultrasensitivity For a genetic toggle switch with two mutually repressing proteins such as the classic switch built by Gardner et al. On the other hand this phenomenon actually provides a way of making artificial switches tunable. It is possible to engineer a biased switch merely by adding a load on the opposite side of the toggle, which is a useful device when engineering a switch that is designed to be switched on only in special circumstances. A load on both repressor proteins similarly stabilizes both sides of the toggle switch. This could be useful when working with synthetic components with low concentrations in cells, especially those that display stochastic switching. A load on both repressor proteins can significantly increase the stability of such a toggle.In natural systems, mutually repressing toggle switches are often found with other complexities, such as a positive feedback motif on one side. The positive feedback motif by itself biases the toggle switch by stabilizing the side it is on at the expense of the other side. A load on the same side then stabilizes the opposite side, and can re-establish balance between the two quasi-potential wells. For engineering circuits in multi-cellular organisms, it is worth noting that that feedback between the load on a toggle switch and the strength of the positive feedback may ensure that the switch operates efficiently even in the presence of cell to cell variability in the load. How loads vary between cells and in multi-cellular organisms is an interesting question to explore in future work. The presence of the positive feedback provides a potential target for evolutionary fine-tuning of the switch.In the above analyses we use novel potential landscape methods that have proved useful and insightful in fields such as protein folding to discuss the fundamental properties of a dynamical system that shows not apparent changes in its stability properties. We demonstrate that these methods, though still relatively underdeveloped for use with non-equilibrium chemical reaction systems, hold promise for understanding the dynamics of such systems beyond what linear stability analysis can provide. However there are certain conditions when addition of a load changes the stability properties of the genetic toggle switch. One class of such effects happen when the repressor can leak away from the repressor-load complex, as can happen either when the repressor can decay or degrade when bound to the load, or when the load can modify the repressor and make it unable to repress. We show, employing standard bifurcation analysis, that additional loads in this system can abrogate the switch-like properties of the toggle switch entirely.In switches based on autocatalysis or positive feedback with an enzymatic deactivation, such as is often found in signaling systems, the effects of a load are equally dramatic. We show that in a simple model of Ras activation, adding a small concentration of Raf molecules changes the bifurcation diagram of the signaling circuit and can completely abrogate the bistability in the system. While we have chosen a specific example of Ras activation, our simplified model, with an autocatalytic forward reaction and an enzymatic backward reaction is a minimal model for a many positive feedback switches. The change in the bifurcation diagram arises from the conservation laws that couple the concentration of the load with the concentrations of the proteins in the upstream module. Given the sensitivity of non-linear dynamical systems to initial conditions, it should probably be expected that many, if not all, positive feedback based switches that operate at the short timescales of signal transduction, and therefore must possess these conservation laws, should exhibit this sensitivity to the effect of a load.Our results throw up an interesting puzzle for quantitative biologists. In many natural signal transduction systems such as the MAPK cascade, the concentration of the output of a bistable switch is quite comparable to the concentration of the load, thus significant changes in load concentrations could have dramatic effects on the behavior of the switch. However it has also been shown that there is a significant cell to cell variability in protein concentrations Previous discussions of the effect of loads on the operation of circuits have suggested the use of insulators, that is circuit elements that insulate the upstream module from the downstream module As can be seen, any additional step or series of steps that can amplify a weak signal can act as an insulator. Another standard example of an amplifying circuit is a phosphorylation cascade which is especially relevant when considering Ras activation since it directly leads to the MAPK phosphorylation cascade. Phosphorylation cascades are also very fast, and therefore do not face the additional time delays of an additional transcriptional step. From the point of view of synthetic circuit design, the insulating mechanism here could be constructed by designing a weak binding affinity of Ras (or the synthetic protein that plays that role) for Raf . The bound complex then catalyzes a phosphorylation cascade that ends by connecting to the downstream circuit.Note that this method of insulation does not have the same time delay costs as the additional transcription steps. However it does come with the metabolic costs of having to produce large amounts of proteins that are essentially serving no useful physiological purpose for the cell. This cost could be relevant in some synthetic biology applications, and certainly needs to be evaluated during circuit design. It has been shown in the context of phosphorylation cycles that insulation always carries a metabolic cost, and in general better insulation carries a greater metabolic cost The existence of the MAPK phosphorylation cascade however begs the question whether it serves the purpose of insulation of the upstream Ras circuit from the downstream circuit. While it is not possible to answer this intriguing question without further experiments, it does appear that the Ras-Raf complex is present is quite large numbers on activated cells. This would suggest that insulation is not the function for which the cascade may have evolved. Our own analysis of the genetic toggle switch with the positive feedback motif suggests that Nature may prefer more complicated forms of regulation that balance the different components of the circuit. However there is no reason why both methods cannot be utilized. To our mind this is a very exciting question that requires more attention from experimentalists and theorists alike.It should also be noted that due to non-specific binding of transcription factors with DNA as well as between proteins, every circuit in the cell, real or synthetic, operates in the presence of a load. Variability in the functioning of circuits that are seen when transferring synthetic circuits between species, or even in different cells, may be a result of not only differences in basic protein concentrations, but also of this undervalued but nevertheless tangible load. Based on this reasoning we predict that some of the host-dependent effects that complicate synthetic biology, i.e. a synthetic circuit that works in one organism not performing well in another, are in fact due to changes in the intrinsic load due to non-specific binding when changing hosts.Our analysis underscores the importance of incorporating loads when simulating models of switches in natural and synthetic systems. Mathematical analysis of switch-like motifs therefore would do well to at least include a load on their output proteins, in order to incorporate the possible effects of load induced modulation on the circuit.Figure S1Surface plots showing response times of the simple genetic toggle switch with changes in load (L) and changes in the dissociation constant (Kd) of binding with load. The units of L and Kd are (molecules/µm3). The z-axis measures the response time indicated in the title. “Same Side Rise” and “Same Side Decay” refers to the rise time and decay time when the load is on the same side as the repressor whose concentration is increasing. “Opposite Side Rise” and “Opposite Side Decay” refers to the rise time and the decay time when the load is on the other side of the repressor whose concentration is increasing. “Both Sides Rise” or decay refer to the rise and decay times when a load is present on both sides . The plot shows that at every Kd, the relation between the response time and load is approximately linear. The response time is largest for the case of “Both Sides Rise” followed by “Opposite Side Rise”. The response time is also non-monotonic with respect to the Kd for a given load, and is maximized at intermediate values of Kd.(TIF)Click here for additional data file.Figure S2Time plot of switching of the simple toggle switch with a load on Repressor 1, at three different values of the dissociation constant. In all three cases the system is switched by providing 150 molecules/µm3 of an inducer at 1000 minutes. The inducer stays constant at that value and is not shown in the plots. The left panel has a very high dissociation constant (Kd = 1000 molecules/µm3) of binding between the load and the repressor, due to which the load has a minimal effect on the system. The middle panel has an intermediate value (Kd = 1 molecules/µm3) because of which the load acts as a dynamic sink by releasing Repressor 1 and slowing the switching. The right panel shows the effect of a small dissociation constant (Kd = 10−3 molecules/µm3). At such strong binding affinities, all of the load is always bound to Repressor 1. Thus the load has minimal effect on the switching dynamics. In all cases total load concentration is 100 molecules/µm3.(TIF)Click here for additional data file.Figure S3Effects of a dynamic load on dynamics of a symmetric toggle switch. (A). The time taken to reach 90% of maximum value for the protein undergoing a low-to-high transition as a function of the equilibrium constant of a dynamic load. Normalized time is a unit-less number defined by the transition time (rise or decay) of the system at a given loading condition divided by the transition time (rise or decay) of an unloaded system. (B). The time taken for the concentration of the protein undergoing a high-to-low transition to reach 10% of its maximum value.(TIF)Click here for additional data file.Figure S4Stochastic time trace and the probability distribution function of repressor concentrations for the large volume simulations. (A). Comparison of time traces of the stochastic simulations of the simple toggle switch with basal parameters (top panel) and a larger volume (bottom panel). The average molecule number is about 5 times greater, and the number of transitions are significantly fewer. (B). The probability distribution function of the genetic toggle switch with the larger molecular number without (left) and with (right) a load. The effect of a load on R1 is qualitatively the same for this system as for the smaller system. Since transitions are slower the data are more uneven for this simulation.(TIF)Click here for additional data file.Figure S5Transition times in a genetic toggle switch with a positive feedback moiety. In all cases the strength of the positive feedback (denoted here by P instead of ρ) is 3.5 on either Repressor 1 (R1) or Repressor 2 (R2). Top Left: Rise time - time to transition INTO state R1 with the positive feedback on R1. Note that the rise time is larger at nonzero loads when the load is on R2 or when the load is on both sides, in agreement with the simple toggle switch. Top Right: Rise time - time to transition INTO state R1 with the positive feedback on R2. Bottom Left: Decay time - time to transition OUT OF state R1 with the positive feedback on R1. Bottom Right: Decay time - time to transition OUT OF state R1 with the positive feedback on R2.(TIF)Click here for additional data file.Figure S6Probability distribution functions of repressor concentrations for the toggle with a positive feedback moiety. The left panel shows that when ρ = 0, the switch is balanced evenly. As ρ increases, the side of the switch with the positive feedback becomes more and more prominent, at the expense of the other side. When ρ = 5, the system spends most of its time in one state.(TIF)Click here for additional data file.Figure S7Bifurcation diagram of the Ras switch with different levels of Raf (load) on the system for the model with Pseudo Steady State Assumption (PSSA). The total number of SOS in the simulation box is used as the parameter being tuned, which varies from 0 to 1000. For Raf = 0, Raf = 10 and Raf = 30, there are two bifurcations points as SOS is increased. In the first bifurcation a new high valued stable steady state appears along with the low valued stable steady state. In the second bifurcation, the low valued stable state disappears leaving behind only the high valued state. The dotted line marks the unstable steady state that also comes into existence in the bistable region. As total Raf increases, the two bifurcations approach each other. When Raf = 50, the system has lost both of its bifurcations and is characterized by a single stable steady state at all values of Raf.(TIF)Click here for additional data file.Figure S8Bifurcation diagram of the Ras activation model based on Law of Mass Action (LMA). Here the total number of Raf molecules (RafT) is the primary parameter being varied. Without Raf, the Ras activation system is bistable as reported. With increasing RafT, the “high” stable steady state branch comes closer with the unstable steady state branch and both are eliminated after a threshold of RafT. A monostable region is maintained beyond the threshold.(TIF)Click here for additional data file.Figure S9T) as the primary parameter.Bifurcation diagram of the Ras activation PSSA model with total number of Raf molecules Click here for additional data file.Figure S10off6.Parameter sensitivity of the bistability of Ras switch to changes in k Increase of koff6 results in leftward shifts of both stable fixed points, increase in the bistable regime and increase in maximal RasGTP activation level (Green Line) when compared to baseline with original value (Blue Line). Decrease of koff6 (Red Line) results in right shift of both limit points, increase in unstable bistable regime and decrease in maximal RasGTP activation level. Qualitative features of bistability are maintained.(TIF)Click here for additional data file.Figure S11on6.Parameter sensitivity of the bistability of Ras switch to changes in k Increase of kon6 (Green Line) results in right shift of both limit points, increase in unstable bistable regime and decrease in maximal RasGTP activation level when compared to baseline original value (Blue Line). Decrease of kon6 (Red Line) results in left shifts of both limit points, increase in bistable regime and increase in maximal RasGTP activation level. Qualitative features of bistability are maintained.(TIF)Click here for additional data file.Figure S12Comparison between bifurcation diagrams of toy genetic toggle switch with and without protection of repressor degradation when bound with promoters. If the protection is not included (Blue Line), a minor increase in the bistable region can be observed with right shift of upper limit point and left shift of lower limit point compared to the case with protection assumed (Red Line). Note that this is not the same as degradation after being bound with the load.(TIF)Click here for additional data file.Figure S13Bifurcation diagram of the genetic toggle switch with positive feedback loop on one side after removal of the protection assumption. The left panel shows the bifurcation diagram when the load is added symmetrically to both sides. Without load molecule, the toggle switch is bistable as predicted. With the increase in LT, the unstable steady state and the “low” stable steady state come closer and meet at certain threshold. The value of “high” stable steady state decreases with increase in LT. Beyond the threshold, the toggle switch becomes monostable. The right panel shows the effect of just adding a load to R1. In this case the high state of R1 approaches the unstable steady state, and annihilates itself. The system jumps to the low stable state, which is equivalent to the “high” state of the other repressor.(TIF)Click here for additional data file.Figure S14Bifurcation diagram of the Ras activation model when Ras can degrade when bound with Raf. As the number of Raf molecules increase, the bistable region decreases. However unlike the case with no protection, the curve moves to the left. When Raf molecules increase by a large amount, bistability is abrogated.(TIF)Click here for additional data file.Figure S15on and k′off values plotted as a function of load for the basic toggle switch.Transition times for various k′ Even if the binding-unbinding rates are slower or much faster than protein decay rates, the load-transition time relationship stays linear. A, C, E, G, I, K, M, O, Q and S show the rise time. B, D, F, H, J, L, N, P, R, and T show decay time.  k′on = 4, k′off = 0.5, Kd = 0.125.  k′on = 10, k′off = 0.5, Kd = 0.05.  k′on = 4, k′off = 4, Kd = 1.  k′on = 10, k′off = 10, Kd = 1.  k′on = 4, k′off = 20, Kd = 5.  k′on = 10, k′off = 50, Kd = 5.  k′on = 4, k′off = 40, Kd = 10.  k′on = 10, k′off = 100, Kd = 10.  k′on = 40, k′off = 400, Kd = 10.  k′on = 100, k′off = 1000, Kd = 10.(TIF)Click here for additional data file.Figure S16on and k′off for the basic toggle switch.Probability distributions of repressor concentrations for various values of k′ Even when the binding-unbinding with the load is several times faster than protein decay rates, the basic phenomena discussed in the paper remains unchanged. (A) k′on = 50, k′off = 500 (B) k′on = 500 k′off = 500 (C) k′on = 500 k′off = 5000.(TIF)Click here for additional data file.Figure S17Bistability of the toggle switch with positive feedback. A bifurcation diagram of the simple toggle switch with a positive feedback moiety on one side, with respect to the parameter ρ that measures the strength of the positive feedback. Only the concentration of R1 is shown for simplicity. The switch remains bistable till ρ becomes larger than a little over 200.(TIF)Click here for additional data file.Table S1off, Kon and β.Slopes of linear fits to rise and decay time with various values of K The first column reports the values of the dissociation constant (Kd = Koff/Kon) and the kinetic constants of the binding of Repressor 1, 2 or the value for β, which represents promoter strength. The other columns report the slopes of the linear fits of the various rise times and decay times. In all cases the fits have high R-squared values (>0.95). Intercept is 1, as the slopes are normalized to the un-loaded transition time. For Kd we change the parameters by two orders of magnitude in both directions to show that the linear relation is robust despite these changes. Note that the relation between rise time or decay time and the binding constant is non-monotonic. Units are as reported in the text.(DOC)Click here for additional data file.Table S2Exponential Fits of the amount of inducer required to transition states as a function of load. The basic genetic toggle switch switch was toggled to its other state by production of the other repressor protein by an inducer, given here as a bolus with a decay rate as shown. The size of the bolus was increased until the state changed. This was repeated at different levels of load and the minimum size of the bolus required was fit by an exponential function of the load. The fits are shown here, along with their R-squared values. “Load applied to the opposite side” means switching from a state without a load to a state with a load. “Load applied to the same side” means switching from a state with a load to a state without a load.(DOC)Click here for additional data file.Table S3Exponential Fits of the amount of inducer required to transition states as a function of load, in the case of induction by repression. The switch was toggled to its other state by repression of the current state by an external molecule, given to the system as a bolus with a decay rate as shown. The size of the bolus was increased until the state changed. This was repeated at different levels of load and the minimum size of the bolus required was fit by an exponential function of the load. The fits are shown here, along with their R-squared value. Thus the inducer required depends exponentially on the load in both the methods of induction. “Load applied to the opposite side” means switching from a state without a load to a state with a load. “Load applied to the same side” means switching from a state with a load to a state without a load.(DOC)Click here for additional data file.Table S4d, load binding rates Kon and Koff, and constant K1.Slopes of linear fits to rise and decay time with a dynamic load, with varying values of load decay rate K The first four columns report the values of the various parameters. The other columns report the slopes of the linear fits of the various rise times and decay times. In most cases the fits have high R-squared values (>0.95). The two exceptions are >0.90 and starred. Intercept is 1, as the slopes are normalized to the un-loaded transition time. Note that for all cases, the relationship between load (expressed here as Keq = Kb/Kd) and transition time is a positive linear relationship.(DOC)Click here for additional data file.Table S5Rate expressions used for the stochastic simulations of the genetic toggle switch. The rate expressions used for the stochastic simulation of the toggle switch along with the description of the reaction are listed.(DOC)Click here for additional data file.Table S6List of reactions in the minimal model of Ras activation. The reactions in the minimal model of Ras activation, along with the labels of the corresponding rate constants are shown. Parameters used in the simulations are given in (DOCX)Click here for additional data file.Table S7Kinetic rate parameters used for the simulations of the Ras model. Here the numbers in the subscript of the rate constants in the “Constant” column refer to the reactions shown in the corresponding row of Supplementary on refers to the on-rate, koff is the off rate and kcat is the catalytic rate. The sources for the rates are as shown in the last column.(DOC)Click here for additional data file.Table S8List of reactions in the toy model of genetic toggle switch. The reactions in the toy model of the genetic toggle switch, discussed in Supplementary (DOCX)Click here for additional data file.Text S1Supporting Information including derivation and analysis of toggle switch and Ras models, parameters used for the simulation and their sources, results of the parameter sensitivity analysis, details of the effect of a dynamic load on the genetic toggle switch, results for the Ras model with Michealis-Menton kinetics, results for the models without the protection assumption.(PDF)Click here for additional data file.


The Standardised Mortality Ratio (SMR) is increasingly used to compare the performance of different healthcare providers. However, it has long been known that differences in the populations of the providers can cause biased results when directly comparing two SMRs. This is potentially a particular problem in neonatal medicine where units provide different levels of care.Using data from The Neonatal Survey (TNS), babies born at 24 to 31 weeks gestational age from 2002 to 2011 and admitted to one of 11 UK neonatal units were identified. Risk-adjusted SMRs were calculated for each unit using a previously published model to estimate the expected number of deaths. The model parameters were then re-estimated based on data from each individual neonatal unit (“reference” unit) and these then applied to each of the other units to estimate the number of deaths each unit would have observed if they had the same underlying mortality rates as each of the “reference” hospitals. The ratios of the SMRs were then calculated under the assumption of identical risk-specific probabilities of death.7243 babies were included in all analyses. When comparing between Network Neonatal Units (Level 3) the ratio of SMRs ranged from 0.92 to 1.00 and for the comparisons within Local Neonatal Units (Level 2) ranged from 0.79 to 1.56. However when comparing between neonatal units providing different levels of care ratios up to 1.68 were observed.If the populations of healthcare providers differ considerably then it is likely that bias will be an issue when directly comparing SMRs. In neonatal care, the comparison of Network Neonatal Units is likely to be useful but caution is required when comparing Local Neonatal Units or between units of different types. Tools to quantify the likely bias are required. For at least 150 years The need to adequately adjust outcomes for differences in case mix (risk-adjustment) is well documented is only a ‘single-pair’ method, and if it is applied to a number of groups it may only be thanks to the mercy of Providence that it is not grossly misleading” The SMR is, therefore, a measure of how the outcomes for an individual healthcare provider compared with those of a reference population: e.g. how Unit A's patients did compared with how they would be expected to have done if Unit A had performed at national rates. It has long been recognised that directly comparing the SMRs of two different healthcare providers may be inappropriate as the SMRs for two providers performing equally for each patient type will not necessarily take the same value if the providers' population structures are different Special Care Units (Level 1) which provide special care but do not aim to provide any continuing high dependency or intensive care, Local Neonatal Units (Level 2) which provide high dependency care and some short-term intensive care as agreed within the network and at least one Network Neonatal Unit (Level 3) which provides the whole range of medical neonatal care  This bias might raise particular problems in the direct comparison of SMRs where there are large differences in the types of patient treated in the healthcare providers. This is seen in neonatal care where in the United Kingdom (UK) neonatal units are organised into networks with different units providing different levels of care In this paper we will describe the source of this potential bias and will quantify the bias which may arise when comparing SMRs for in-unit deaths of very preterm babies  admitted for neonatal care in the UK. We were unable to find any previous examples of the potential size of this bias in any clinical specialty. A suggested method for quantifying the differences in case-mix between two populations, the M-statistic, will be examined to investigate the size of the bias and the value of this statistic.The Neonatal Survey has been given permission to collect data by the Patient Information Advisory Group .O to the number expected E calculated using the proportion of events occurring in a reference population: i.e.The SMR is the indirectly standardised ratio of the observed number of events Aiπ is the probability that the event will occur for an observation in the ith case-mix stratum in Provider A; Riπ is the probability that the event will occur for an observation in the ith case-mix stratum in the reference population; Aip is the proportion of observations in the ith case-mix stratum in Provider A.where Therefore, for the SMRs for two providers (Provider A & Provider B) to be equal the following would need to be true:Aiπ = Biπ = ABiπ for all values of i), and different from the reference population for at least one stratum, the SMRs would only be sure to take the same value if their population structures were also the same :However, even if the stratum-specific event probabilities were identical for both providers for all strata . The size of any difference between the SMRs of two providers with the same stratum-specific event probabilities depends on the size of the difference between the risk-specific mortality in the units (ABiπ) and that of the reference population (Riπ) and on the size of difference in population structure between the units (Aip and Bip).In other words, even if two healthcare providers were performing identically for each type of patient, their SMRs would not be the same value if the proportion of each patient type was different for each provider , a population-based audit of in-patient neonatal care based in the East Midlands and Yorkshire Regions of the UK Data were extracted for all preterm babies admitted to neonatal care who were born from 2002–2011 at 24 to 31 completed weeks gestational age. Observations were excluded for missing or implausible birthweight; missing or ambiguous gender and inevitably lethal congenital anomalies.Approximately 28% of very preterm babies recorded in TNS are transferred between neonatal units during their period of neonatal care. In this paper the ‘unit of care’ was defined as the unit that the baby was a patient in at the end of their third day of life or the unit of death if the baby died before the end of the third day of life. It is expected that by the third day of life a baby's level of risk will have been fully assessed by the clinician team, the baby will have been stabilised and transferred to the unit which can provide the most appropriate care.An SMR was calculated for each neonatal unit by dividing the observed number of deaths by the expected number of deaths estimated using a published mortality model and parameters estimates derived using TNS data A 95% confidence interval was calculated using the normal method In order to estimate the likely bias that would arise solely from the differences in case-mix when directly comparing SMRs from two different neonatal units, pair-wise comparisons were made assuming that the observed risk-specific mortality was identical for the two units. This was done by using the data of each unit in turn to calculate new ‘reference’ unit specific estimates for the parameters of the model, i.e. for Unit A:These estimates were then applied to each of the other units to estimate the number of deaths that would have been observed if their risk-specific mortality was the same as the ‘reference’ unit and an SMR was then calculated for each of the other units. Any difference between these SMRs must be due to the differences in the units' population structures and the ratio of the SMRs was calculated to quantify the size of this difference. Therefore applying the observed mortality probabilities from unit A to unit B gives the ratioSince the observed number of events for each unit was estimated using the same case-specific mortality for each of the pair-wise comparisons, the SMRs for the two neonatal units would be expected to be equal if there were no differences in the case-mix structure.et aliπ≥0.96; 0.91≤iπ<0.96; 0.76≤iπ<0.91; 0.51≤iπ<0.76; 0.26≤iπ<0.51; iπ<0.26.The M-statistic is a measure of agreement between two populations and was calculated in this analysis using the methods, and ranges, outlined by Boyd The M-statistic of any particular pair of units is the sum of the lower of the two units' proportions in each range:Arp and Brp are the proportion of babies in range r for units A and B respectively.where An M-statistic of 1 is achieved if the proportions are identical for every range, the lower the M-statistic the more divergent the populations. The M-statistic was calculated for each pair-wise comparison and then plotted against the ratio of SMRs to examine the relationship between each pair of units.In total 7,340 eligible babies were identified from TNS. Babies were excluded for lethal congenital anomalies ; missing or implausible birthweight ; missing or indeterminate gender . Therefore, 7,243 babies were included in the analyses.The Neonatal Network Units admitted more babies than the Local Neonatal Units, and these babies tended to be born at earlier gestational ages, have lower mean birthweight and were more likely to die before discharge. The Local Neonatal Units varied widely in the number of babies admitted and the proportion of deaths . The estThe values of the ratios of SMRs obtained when assuming the risk-specific mortality probabilities are equal in pairs of units are given in Iα = 9.92; Iβ = −0.44; Iγ = 0.60; Iδ = −0.28 and, therefore, the bias that arose when this model was applied was very small. On the other hand, comparisons using the estimates obtained using Unit D's data produced large values for the ratio of SMRs (1.11 to 1.68) because the model estimates were very different to the published estimates: Dα = 11.59; Dβ = −0.56; Dγ = 1.64; Dδ = −0.11.The values for ratios obtained by applying the risk-specific probabilities from Unit I to other units were particularly close to 1 (0.99 to 1.01). The estimates for the model parameters obtained using data from Unit I alone were similar to values in the full model: M-statistics were then calculated for each pair of units and were plotted against their corresponding ratio of SMRs . There wIn this paper the potential bias associated with using SMRs to directly compare case-mix adjusted mortality of very preterm babies between two neonatal units has been described and quantified. This problem is a form of Simpson's Paradox The observed bias tended to be small when comparing Network Neonatal Units as their populations were more similar than when comparing across all units, since they admit similar proportions of high risk babies and have generally similar patterns of transfers. In principle, pairs of units with a high value for the M-statistic would be expected to have values for the ratio of SMRs closer to the value 1, as units with very high M-statistics had similar proportions of babies at the different levels of risk. However, in this research the M-statistic alone did not appear to be a reliable diagnostic tool for predicting the bias. Even for values of the M-statistic above 0.9, five pairs of SMRs showed differences of over 20% . This waAlternative summary statistics have been suggested which, unlike the SMR, do allow direct comparison between two healthcare providers. The Comparative Mortality Figure (CMF) is obtained by direct standardisation; the ratio of the number of events observed in the reference population to the number expected calculated using the risk-specific probability of the event for the healthcare provider In practice this research shows that when comparing neonatal survival of preterm babies the direct comparison of SMRs from units providing different levels of care could lead to misleading conclusions. We have only investigated the potential bias under the assumption of no difference in the risk-specific mortality probabilities between units. In reality, these probabilities will differ between units but this bias could alter, and perhaps reverse, these real differences in the SMRs. It is important when directly comparing SMRs for neonatal mortality that comparisons between units of different types should be undertaken with particular caution. The same caution should be applied when directly comparing between Local Neonatal Units as these also appear to differ in case-mix sufficiently to introduce bias to direct comparisons. The size of the bias that would lead to any comparison being misleading obviously depends on the purpose for which the comparison is being made. However, it is likely that observed differences in SMRs that are 20% greater than the ‘true’ difference would need to be interpreted with caution.The main limitation of this work is that the estimates for the ‘reference’ unit specific model parameters for Local Neonatal Units were derived from small samples with few deaths. This means that it is possible that the estimated risk-specific probabilities of death from these models could be implausible for some groups of babies, especially where the model was extrapolated beyond observed data. Since the size of the potential bias present when comparing two SMRs is dependent on the difference between the risk-specific probabilities of death in the reference group and those observed in the unit of interest, any extreme estimates derived for the Local Neonatal Units may have resulted in overestimates of the likely values of the bias found in practice. This may be the case with the model developed from Unit D, although the values of the parameter estimates appeared plausible on inspection.This work is based on a relatively small number of neonatal units and the results obtained may not necessarily be generalizable to other neonatal units. However, there is no evidence to suggest that the organisation and practice of these units is different from that of other UK neonatal units.If the populations of healthcare providers differ considerably then it is likely that bias will be an issue when directly comparing SMRs. In neonatal care, the comparison of Network Neonatal Units is likely to be appropriate but caution is required when comparing between Local Neonatal Units as these differ considerably in case-mix from each other. More generally it is important that there is a proper understanding of whether a single risk model should be used for comparing any group of service providers.Future work is required to develop a better tool than the M-statistic to quantify the likely bias.