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- Methodology article
- Open Access
Transforming Boolean models to continuous models: methodology and application to T-cell receptor signaling
© Wittmann et al; licensee BioMed Central Ltd. 2009
- Received: 19 January 2009
- Accepted: 28 September 2009
- Published: 28 September 2009
The understanding of regulatory and signaling networks has long been a core objective in Systems Biology. Knowledge about these networks is mainly of qualitative nature, which allows the construction of Boolean models, where the state of a component is either 'off' or 'on'. While often able to capture the essential behavior of a network, these models can never reproduce detailed time courses of concentration levels.
Nowadays however, experiments yield more and more quantitative data. An obvious question therefore is how qualitative models can be used to explain and predict the outcome of these experiments.
In this contribution we present a canonical way of transforming Boolean into continuous models, where the use of multivariate polynomial interpolation allows transformation of logic operations into a system of ordinary differential equations (ODE). The method is standardized and can readily be applied to large networks. Other, more limited approaches to this task are briefly reviewed and compared. Moreover, we discuss and generalize existing theoretical results on the relation between Boolean and continuous models. As a test case a logical model is transformed into an extensive continuous ODE model describing the activation of T-cells. We discuss how parameters for this model can be determined such that quantitative experimental results are explained and predicted, including time-courses for multiple ligand concentrations and binding affinities of different ligands. This shows that from the continuous model we may obtain biological insights not evident from the discrete one.
The presented approach will facilitate the interaction between modeling and experiments. Moreover, it provides a straightforward way to apply quantitative analysis methods to qualitatively described systems.
- Continuous Model
- Boolean Function
- Additional Data File
- Boolean Network
- Ordinary Differential Equation
Close interaction between experiments and mathematical models has proven to be a powerful research approach in Systems Biology. Especially the modeling of regulatory and signaling networks, however, is typically hampered by a lack of information about mechanistic details, as often one can only determine the interactions of the involved species in a qualitative way. The current shift of focus in Systems Biology from single signal transduction pathways to networks of pathways exacerbates this lack of information even more. Therefore, the creation of mass action based models that accurately describe the underlying biochemistry is typically restricted to small well-studied subsystems.
Boolean models are a very coarse description of biochemical processes. They phenomenologically describe observed dependencies often leaving out still unknown players or intermediate steps. As our transformation requires no additional information the resulting continuous models are, of course, still phenomenological models. We can automatically create these continuous phenomenological models out of a Boolean model, but we cannot create a mass action law without additional knowledge on the biochemistry. Our method is a top-down approach for (large) networks with incomplete mechanistic knowledge — derived e.g. from pathway databases — where predictive kinetic modeling is infeasible. The main point that we want to make in this contribution is that also these phenomenological models can be used in Systems Biology to explain and predict quantitative experimental results.
scenario 1, resting state: All inputs are set to 0, feedback loops are deactivated.
scenario 2, early events: Inputs are set to 1, feedback loops are still deactivated.
scenario 3, mid-time events: Inputs are still 1, feedback loops are active.
A qualitative analysis of an expanded version of this Boolean model by Saez-Rodriguez et al. [8] yielded new and non-obvious signaling pathways. There are also quantitative models covering aspects of T-cell signaling in mechanistic detail, e.g. [9]. In contrast thereto, the continuous model we obtain in the following describes the T-cell signaling cascade on a larger scale yet in less mechanistic detail.
We apply the transformation method to the T-cell model. The resulting continuous model is able to fully reproduce the behavior of the Boolean model. Moreover, we show that it can easily include and deal with the different time scales of interactions in the three scenarios. Hence the continuous model is indeed a generalization of the Boolean model with richer dynamic properties. This is in line with previous findings [10, 11], indicating that the qualitative behavior of a discrete model is reproduced by its continuous homologue. We can further corroborate this hypothesis by generalizing existing theoretical results on the steady-states of discrete and continuous models.
The crucial question is, of course, whether the continuous model derived from a Boolean network is indeed competent to explain an aspect of biological reality in a precise quantitative fashion. We answer this question in the affirmative by showing that our T-cell model reproduces time courses of concentration levels measured for three different ligand concentrations [12]. Moreover, it is able to predict binding affinities of different ligands from their induced signaling profiles. The fact that the model can differentiate between more than two different concentrations shows that we have definitely left the Boolean (binary) world behind.
Representation of Boolean functions and models
A Boolean model consists of
giving the value of x i at the next time step for every possible combination of .
This is a so-called sum-of-product representation of B i . These representations are especially convenient, as they allow to graphically represent our models using interaction hypergraphs [6]. The idea is to represent each product (AND gate) in the sum-of-product form of B i by a directed hyperedge between a set of start nodes and the end node X i . Each pair (s, X i ), s ∈ S, carries a sign — '+' or '-' — depending on whether there is a factor s or ¬s in the product. All incoming hyperedges at node X i are then a graphical representation of B i . This is further illustrated in Additional data file 1.
The general approach to making discrete models continuous
The first step for obtaining a continuous model from a Boolean one is to replace the discrete variables x i by continuous variables taking values in [0, 1], i.e. we normalize concentrations to the unit interval. Consequently, we have to 'extend' the functions B i to functions : [0, 1] N → [0, 1]. We call the functions continuous homologues of the Boolean functions B i . The crucial point is, how the transformation of a Boolean function into a continuous homologue is performed. We will address this issue in the next section and continue with the outline of the general approach.
In numeric simulations the discretization of time is obviously irrelevant. It complicates, however, the detection of small-scale continuous effects and is a serious drawback in the further investigation of the model by analytical methods.
and by setting τ i = 1/γ i and we obtain the ODEs (2).
Herein we focus on model (2), as this model can be further analyzed using the rich and mathematically rigorous theory of ODEs. Note furthermore that model (1) can be considered a special case of model (2) after numeric integration.
Continuous homologues of Boolean functions
As already mentioned, the key point is how a continuous homologue can be obtained from a Boolean function B i in a computationally efficient manner. A suitable transformation has to satisfy three conditions:
Accuracy: It has to be accurate, which means that and B i must agree on the vertices of the unit cube, i.e. for values in . Functions satisfying this condition are called perfect continuous homologues. As will be shown, for perfect continuous homologues discrete and continuous models exhibit a similar steady-state behavior.
Good analytical properties: The functions should have good analytical properties such as smoothness in order to allow and facilitate a mathematical analysis of the system of ODEs.
Minimality and uniqueness: The functions should be the unique minimal solution in their interpolation class.
The three transformations we propose in the following are all based on multivariate polynomial interpolation [13, 14] (see Methods). Here is defined as a polynomial in the variables that agrees with B i on the vertices of the unit cube. As will be shown in the Methods section, this technique satisfies all three of the above requirements. There are other approaches which we shortly review and compare in the Discussion section.
by linearly interpolating the functions B i using the technique of multivariate polynomial interpolation as explained in the Methods section. These functions are called BooleCubes. By substituting for B i in equation (2), we can then define a system of ODEs that describes the temporal development of the .
The functions are affine multilinear, i.e. for each 1 ≤ j ≤ N i and fixed , k ≠ j, there exist constants a, b ∈ ℝ such that . Molecular interactions, however, are known to show a switch-like behavior, which can be modeled using sigmoid shaped Hill functions [15] (see Additional data file 2). The two parameters n and k have a clear biological meaning. The Hill coefficient n determines the slope of the curve and is a measure of the cooperativity of the interaction. The parameter k corresponds to the threshold in the Boolean model, above which one defines the state of a species as 'on'. Mathematically speaking, it is the value at which the activation is half maximal.
which we call HillCubes. Plots of the HillCubes of all 16 two-variable Boolean gates can be found in Additional data file 3. Now a new system of ODEs can be defined by replacing the in equation (2) by the HillCubes .
normalized HillCube
Note that Hill functions never assume the value 1, but approach it asymptotically. Hence, the HillCubes are not perfect homologues of the Boolean update functions B i . If this is desired a simple solution is to normalize the Hill functions to the unit interval. This yields another (perfect) continuous homologue of the
which we call normalized HillCubes, and thus defines also a new continuous ODE model.
Theoretical results about the relation between discrete and continuous models
A natural and interesting question is how similar the discrete and the continuous model are. It can easily be shown (see Methods and [11]) that, whenever the in equation (2) are perfect homologues of the Boolean update functions B i , a steady-state of the Boolean model will also be a steady-state of the continuous system. This result can be applied to the BooleCube model as well as the normalized HillCube model, but not to the non-normalized HillCube model. Therefore, Boolean steady-states will in general not be steady-states of the non-normalized HillCube system. The question is if Boolean steady-states still correspond to 'similar' steady-states of this continuous model, at least for certain parameters. Using the implicit function theorem we were able to show that this is indeed the case (see Methods). The reverse statement is, of course, not true, as in the continuous model additional (stable) steady-states may arise. Besides the steady-state behavior, monotony properties are a further important characteristic of a dynamical system. These properties determine the effect of a down- or up-regulation of a certain species on the other species. Due to its accuracy the presented transformation method preserves the monotony properties of a Boolean network. In the Discussion section we illustrate this further using the T-cell model as an example.
We show these results as they justify our transformation approach by demonstrating that it preserves essential properties of the Boolean model. For a deeper mathematical investigation of the relation between discrete and continuous models we refer the interested reader to the rich literature on this subject, e.g. [11, 16, 17]
Simulation of the Boolean T-cell model
After deactivation oscillations occurred, artefacts of the synchronous updating. When the species were updated asynchronously according to some permutation, these oscillations could be observed for 3078 out of 10000 randomly sampled permutations; in the other cases a steady-state was reached.
Continuous T-cell model
We applied the transformation technique to the Boolean T-cell model. Using non-normalized HillCubes we obtained a large quantitative model of T-cell activation (see Methods). In a first step, we manually determined approximate parameters for which the continuous model reproduces the behavior of the Boolean model during all three scenarios. The information about fast and slow interactions was thereby encoded in the values of specific parameters. Consequently, the continuous model was able to explain both, the activation as well as the deactivation of T-cells, without any alteration of the network topology between different scenarios. Note that this was necessary in the Boolean case. We set all Hill coefficients to n = 3, the thresholds to k = 0.3 and the life-times to τ = 1, with the exception of Fyn and ZAP-70. Here we knew that the interactions Fyn → PAG-Csk and ZAP-70 → cCbl operate on a slower time scale. Therefore the thresholds for these interactions were set to kslow = 0.8 whereas the thresholds for the interactions Fyn → TCR-phos, ZAP-70 → LAT-phosp, ZAP-70 → PLCg (act) and ZAP-70 → Itk were set to kfast = 0.1. The Hill functions for the three thresholds k, kslow and kfast are displayed in Figure 2B. Finally, we set the life-times of Fyn and ZAP-70 to τFyn = τZAP-70 = 10 to enlarge the time gap between the two switching points.
The numeric simulation of the continuous T-cell model with the manually determined parameters is shown in Figure 3B. At first, the cell was again in its resting state with all three inputs turned off. Then at t = 10 we manually switched on all inputs and the signaling cascade was triggered off showing an expression profile very similar to the Boolean simulation (Figure 3A). We observed a total activation of the four transcription factors at around t = 21 just like in the Boolean simulation. One can clearly see that Fyn and ZAP-70 were activated more slowly. Nonetheless TCR-phos, LAT-phosp, PLCg (act) and Itk were instantly turned on due to the low threshold kfast. Only when at around t = 25 the concentrations of Fyn and ZAP-70 were high enough, the feedback loops Fyn → PAG-Csk and ZAP-70 → cCbl became active and began to switch on PAG-Csk and cCbl. While PAG-Csk reached a constant medium expression level, cCbl was only weakly and transiently expressed. This, however, sufficed to switch off the cascade and ultimately at around t = 67 all transcription factors were again deactivated. In contrast to the Boolean model, the continuous model did not exhibit an oscillatory behavior but reached a steady-state after deactivation of the signaling cascade. The species with long expression periods in the Boolean oscillation (Tcr bind, PAG Csk, Fyn, TCR phos and IkB) were expressed at high or medium levels whereas the species with short expression periods were not expressed at all.
Comparison of the discrete and the continuous T-cell model
Although one can argue that there is no real time scale in Boolean networks, we compared both models by substracting the discrete from the continuous time course (Figure 3C). The activation of the feedback loops in the Boolean model was conveniently chosen such that we have a total deactivation of the transcription factors at around t = 67 in both models. While we observed an almost perfect agreement of the time courses of the three inputs, there was a huge difference in the time courses of the species involved in the two feedback loops. This was not surprising, considering that these loops are regulated differently in both models. The species downstream of the regulatory loops (from LAT phosp to CRE in Figure 3C) showed again a similar expression pattern.
We then analyzed this part of the cascade more deeply. Figure 3C also shows the correlation between the discrete and the continuous expression pattern of the species downstream of the feedback loops at each time point 0 <t < 100. We observed a high correlation in the more stationary phases (resting state, activated state and deactivated state) and a significant drop of correlation during the transitions between these phases. This met our expectation that the two models show the same qualitative but a different dynamic behavior.
Ratio of activation and deactivation time
When looking at Figures 3A and 3B, a striking difference between the dynamics of both models is that in the discrete model activation and deactivation took approximately the same time, whereas in the continuous model activation was a much faster process than deactivation. This can also be seen from the red and blue 'steps' in upper Figure 3C during the activation and deactivation phases. To confirm that this was not merely an artefact of our choice of parameters, we calculated and analyzed the ratio between activation and deactivation time for different parameters. We defined
the beginning of activation tActBeg as the time when the first species (of the lower 29) reaches 5% of its maximal value,
the end of activation tActEnd as the time when the last species reaches 95% of its maximal value,
the beginning of deactivation tDeactBeg as the time after tActEnd when the first species drops under 95% of its maximal value,
the end of deactivation tDeactEnd as the time when the last species drops under 5% of its maximal value,
In the Boolean model we could easily compute ρb = 1 implying equally fast activation and deactivation. In the continuous model the crucial parameters were the life-times τFyn and τZAP-70 on the one hand and the concentration thresholds kfast and kslow on the other hand. The remaining parameters were set to n = 3, k = 0.3 and τ = 1.
First, we computed ρ for fixed kfast and kslow and different τFyn = τZAP-70. The result is shown in Figure 3D. For τFyn = τZAP-70 = 1 the cascade was not activated properly. For larger values we observed a decrease in ρ implying that an increase of τFyn and τZAP-70 prolonged the deactivation phase. This was to be expected — longer life-times resulted in a lessened increase of the decisive elements Fyn and ZAP-70 in the regulatory loop.
Second, we analyzed the effect of kfast and kslow for fixed τFyn = τZAP-70 = 10. The result is shown in Figure 3E. Only for parameters kfast ≪ kslow < 1 the cascade was activated properly. This agrees well with the fact that the difference between these two parameters is responsible for the delayed activation of the feedback loops. If it was not big enough the cascade was being deactivated before it had been fully activated. The greater this difference was, i.e. the farther we go away from the diagonal in Figure 3E, the smaller ρ got, implying that a later activation of the feedback loops prolonged the deactivation phase. However, despite all these influences of the parameters, we observed much smaller ratios ρ in the continuous model than in the Boolean model. The average ρ's in Figures 3D and 3E were = 0.24 and = 0.27, respectively, which were both significantly smaller then ρb = 1. This suggests that ρ ≪ 1 may be an invariant of the dynamical system of biological importance.
Explanation of experimental data using the continuous T-cell model
Best fit parameter set for ligand L144.
ext. sig.
ext. sig.
ext. sig.
Best fit parameter set
Subsequently, we analyzed the distribution of the Hill parameters within the best fit parameter set for ligand L144 (Table 1). When looking at the distribution of the exponents n upstream of the measured species (Figure 4F) we found that after the optimization 36 out of 49 (73%) were below the ad hoc estimate (3) and only 8(16%) were above it. This was to be expected, as in order to fit three different concentration levels the model had to contain mainly slow switches (low n) and only few Boolean-like switches (n → ∞). Figure 4E shows the distribution of the threshold parameters k upstream of the measured species. As explained above, we had manually set these parameters to 0.3, with the exception of four k's which had been set to kslow = 0.8 and two k's which had been set to kfast = 0.1. Interestingly, this structure had been preserved during the optimization. The mean of the thresholds k was 0.298 and hence well agreed with the ad hoc estimate. Also the high and low thresholds were still at least two standard deviations away from the mean, cf. the red and blue markers '+' in Figure 4E. Only one other parameter k also had a Z-score above 2: the threshold kL144 for the stimulation of the TCR by the ligand L144, cf. marker 'o'. Possible implications hereof are discussed below.
Due to the large number of parameters the obtained parameter set was, of course, far from being unique. But we showed that a continuous model inferred from a Boolean model is able to reproduce experimental data in a quantitative way. Moreover, this transformation could enhance the explanatory power of the model in the sense that it was enabled to differentiate between more than two states.
In our example, the threshold parameters k are rather tightly centered around their mean. In principle, however, we could also have extreme outliers, i.e. very large or very small (≈ 0) values in the distribution. Mapped back to the Boolean model, this would imply a change of the network topology, as the corresponding reactions are then quasi-constant, either 'off' (for very large k) or 'on' (for very small k). Thus, a fitting of the continuous model to experimental data may also yield information about the network structure of the Boolean model.
Prediction of binding affinities of different ligands
As already mentioned, the fitted threshold kL144 for the stimulation of the TCR by the ligand L144 was significantly below the mean of the other thresholds k, cf. marker 'o' in Figure 4E. This gave rise to the question of which affinities the model predicts for the other two ligands. To this end, we fitted the affinities of Q144 and Y144, mapped to the inverse of the Hill function's threshold parameters kQ144 and kY144, respectively, keeping the rest of the parameters constant. As expected, the fits themselves were far from perfect, due to parameter indeterminacies. Surprisingly however, the values of kQ144 and kY144 we obtained were significantly above the mean of the other thresholds k, cf. markers '◇' and '□' in Figure 4E. This suggests that the predicted relation between the parameters kL144 ≪ kY144 <kQ144 is not simply an artefact of the optimization process. And indeed, it agrees well with experimental data [21].
We now further discuss the presented transformation method and compare it to various other approaches. Also the relation between discrete and continuous models is discussed, especially with respect to their steady-state behavior and monotony properties.
Comparison of different transformation approaches
The relation between discrete and continuous models has already been investigated and various approaches to the problem of constructing the continuous homologues of the Boolean update rules have been proposed. In the following we shortly review previous work and compare the different approaches.
Piecewise linear differential equations
The idea to compare continuous and discrete models is almost as old as Boolean modeling itself. In 1973, Glass et al. [11] studied the relation between discrete models and ODE models of the form (2). Their motivation, however, was quite the opposite of ours. While we intend to enrich the dynamic behavior of discrete models, Glass et al. wanted to investigate the qualitative properties of continuous networks by studying corresponding simpler discrete models. They propose Hill functions as a suitable continuous homologue of one-variable Boolean step functions. In the case of multi-variable Boolean functions B i a (perfect) continuous homologue is constructed as follows: Note that, when building a Boolean model, one implicitly introduces a threshold 0 <θ i < 1 for each species X i and defines its state as 'on' if its concentration is above this threshold. The hyperplanes = θ i , i = 1, 2, ..., N decompose the cubes into rectangular regions called domains. Each of these domains contains exactly one vertex and is denoted by accordingly (see Additional data file 4). A simple way of defining the functions is now to set
With this definition model (2) is a so-called piecewise linear ODE model which means that within each domain equation (2) is a linear ODE of the form . This kind of equation is very well understood and can be solved analytically. The functions defined in (6) perfectly agree with B i on the vertices of the unit cube. Using piecewise linear ODEs Glass et al. could prove some theoretical results on the relation between discrete and continuous models, e.g. that Boolean steady-states are also steady-states of the continuous model. Some of these results are restated and generalized in the Methods section. Piecewise linear models are typically not used for quantitative simulations, as the step-like transition between the different domains is often unrealistic. Rather they are analyzed in a qualitative and semi-qualitative way, where their trajectories between the different domains are treated analogously to the state transition graphs of Boolean models [22].
Fuzzy logic
Another well studied way of generalizing Boolean models is fuzzy logic [23]. Recall that in a Boolean model one defines the state of a species as 'on' if its concentration is above a certain threshold. In fuzzy logic this concept is relaxed and a so-called degree of membership (DOM) function is introduced for each species X i . For concentrations 0 = = 1 this function gives the degree with which we say that X i is 'on'. There are two standard ways of generalizing the Boolean operators AND, OR, NOT:
Standardized qualitative dynamical systems
The right-hand side of the above ODE consists of two parts: an activation function and a term for decay as in (2). The activation is given by a sigmoid shaped function of ω i , where ω i represents the total input to node X i . The steepness of the activation function is determined by the parameter h. Decay is assumed to be proportional to . Actually, a more general form of ω i is introduced in [10], where the influence of the activators and inhibitors can be differently weighted. For the sake of better comparability we set all these weights equally to 1, as suggested in [10]. The more activators and the less inhibitors of a node are 1, the greater ω i is. It takes its minimum (0), iff all activators are 0 or all inhibitors are 1, and its maximum (1), iff all activators are 1 and all inhibitors are 0. This, however, does not exactly correspond to the Boolean rule from above. In consequence, steady-states of the Boolean model are not necessarily steady-states of the continuous model. This will be further discussed below.
Multivariate polynomial interpolation
The aforementioned attempts only lead to functions which are either not differentiable or do not precisely generalize the Boolean logic. A straightforward approach to eliminate these drawbacks is to use multivariate polynomial interpolation [13] for the construction of the functions (see Methods). This technique can be applied to any Boolean function B i . The resulting BooleCubes are smooth and can easily be analytically differentiated and integrated. They agree with the Boolean functions B i on the vertices of the unit cube and hence the steady-states from the Boolean model are also steady-states in the continuous model.
The HillCubes do not perfectly agree with the Boolean update functions due to the asymptotic behavior of the Hill functions. By a suitable choice of the Hill parameters the difference can be reduced but not fully eliminated. An easy way to achieve a perfect agreement is to normalize the Hill functions to the unit interval, as is done in the normalized HillCubes from equation (5).
To conclude, we illustrate the above methods applied to a simple OR gate between two species X1 and X2. We compute
from equation (6),
obtained by fuzzy logic (with linear DOM functions) following (7),
obtained by fuzzy logic (with linear DOM functions) following (8),
the input function from equation (9) introduced by Mendoza et al.
the BooleCube from equation (3) obtained by the interpolation technique,
the HillCube from equation (4) for Hill functions f1 and f2 with parameters n = 3, k = 0.5,
of from equation (5).
Figures 5C and 5D show the product-sum fuzzy logic function and the input function ω. One can clearly see that they do not represent a pure OR gate, where the values at (x1, x2) = (1, 0) and (x1, x2) = (0, 1) should already be maximal. This is the case in Figures 5A and 5B which show the piecewise linear and the min-max fuzzy logic function . Here however, the problem is that the functions are not differentiable, as can easily be seen from their plots. The BooleCube shown in Figure 5E is both, smooth and maximal as soon as any concentration is equal to 1. Finally, Figures 5F and 5G show the (normalized) HillCubes and , respectively, which are also smooth and can be considered good transformations of the Boolean OR gate. An overview about the discussed advantages and disadvantages of the different transformation techniques is provided in Figure 5H.
Theoretical results about steady-states
A fundamental principle of biological modeling is that steady-states of a model typically correspond to the different operating modes or states of the biological system under study. This correspondence was also the motivation for Kauffman's seminal study [1], where Boolean models were introduced for the first time in biology. A critical step in the justification of any transformation method therefore is to ensure that at least the steady-states of the Boolean model are still steady-states in the homologue continuous system. In the case of a perfect agreement of the Boolean update rules B i with their continuous homologues this can easily be shown (see Methods and [11]). This perfect agreement, however, is not a biologically plausible assumption; biological interactions, such as enzyme kinetics for example, are known to asymptotically approach but never fully reach a saturation level. Empiric evidence that also in real-world examples, the steady-states of a Boolean model correspond to steady-states of a homologue continuous model, is given by Mendoza et al. [10]. The method of transformation used therein has already been described above and we also mentioned that it does not accurately transform the Boolean update rule into a continuous activation function. This inaccuracy is due to a systematic difference between the Boolean logic and the analytic form of the activation function. It is not the result of an asymptotic sigmoid function; in fact, the used sigmoid function assumes its maximal values 0 and 1. One can easily construct an example where due to this systematic difference Boolean steady-states are not conserved under the transformation (see Additional data file 5).
In the case of the HillCube model, it is the other way round. There is no systematic difference between the Boolean update rules and the HillCube functions, the imperfect agreement is caused by the asymptotic behavior of the Hill functions. Therefore, the difference between both can be made arbitrarily small — albeit not zero — by a suitable choice of parameters. In this situation, we can show that for certain parameters, more precisely for sufficiently large exponents, there will be a steady-state of the continuous system in the neighborhood of each Boolean steady-state (see Methods). This theoretical result further justifies the presented transformation method.
Monotony properties
A nice feature of our method for converting Boolean into continuous models is that monotony properties, typically captured in the underlying interaction graph of the system, are preserved. Interaction graphs are signed directed graphs where each directed edge reflects a causal dependency, which can either be positive or negative, between its start and end node. Boolean models represented as interaction hypergraphs have a unique underlying interaction graph which can easily be derived from the logical model (by splitting the ANDs, see Klamt et al. [6]). For example, the Boolean function A = (¬B ∧ C) ∨ D would be translated into two positive arcs (C → A, D → A) and one negative arc (B ⊣ A). In the interaction graph one may then compute the recently introduced dependency matrix [6, 24], which determines for each pair (X, Y) of species the global effect of X on Y. This effect can — in some cases only initially — be positive, negative, ambivalent or vanishing.
For example, the dependency matrix of the T-cell model tells us, that LAT-phosp exerts purely positive effects on ERK, JNK, IKK, and NFAT, because there are only positive paths from LAT-phosp to these species and no negative feedback is involved. If we simulate a scenario with the logical model and repeat it then with e.g. fixing LAT-phosp to 1 (i.e. to the highest possible value), the resulting Boolean values in the four species mentioned above cannot decrease. In continuous systems, the interaction graph is encoded in the sign structure of the Jacobian matrix. In fact, in a continuous system obtained from a Boolean model the interaction graph is up to negative self-loops identical and monotony properties are therefore preserved. Accordingly, a positive perturbation in LAT-phosp, e.g. by permanently decreasing the threshold of ZAP-70 in the interaction activating LAT-phosp, results in a trajectory that is always above the trajectory of the non-perturbed system (Figure 3F). In fact, in accordance with the interaction graph of the Boolean model, we observe purely positive effects on all species downstream of LAT-phosp with the exception of ikB. Hence, important qualitative properties of the dynamics derived from the logical model are reflected in the dynamics of the continuous system.
With increasing amounts of quantitative data being available, the challenge arises how we can use our typically qualitative knowledge about biological systems to explain this data. For this purpose, we presented a canonical and fully standardized way of transforming qualitative discrete into continuous models. The transformation is accurate and we can show that it preserves the steady-state behavior as well as the monotony properties of the discrete model. The feasibility of the presented approach was substantiated by applying it to a logical model of T-cell receptor signaling. The resulting model is an extensive continuous model of T-cell activation. In contrast to the Boolean model it allowed to accommodate different time scales by adjusting kinetic parameters. It was competent to reproduce time courses of key signaling molecules measured for three different ligand concentration levels. Moreover, the model was able to predict the binding affinities of different ligands.
Being fully automatized [Krumsiek et al.: Odefy — From discrete to continuous models. In preparation (2009)] the presented method recommends itself to be applied to further biological systems. Future work could also aim at generalizing the approach from Boolean (binary) to s-state systems, where one no longer differentiates between two but s > 2 discrete states, e.g. 'low', 'medium' and 'high'. Finally, the relation between a discrete model and its continuous homologue needs to be further investigated, especially with respect to more complex behaviors like oscillations, which are of importance in many biological systems, such as cell cycle.
Multivariate polynomial interpolation
We now explain the technique of multivariate polynomial interpolation of a single Boolean function B i . Therefore i ∈ {1, 2, ..., N} is fixed and for the sake of simplicity the subscript i is omitted. We remark that here B can be any real-valued function on the vertices of the unit cube {0, 1} N , i.e. does not necessarily have to be a Boolean function. The idea is, to find a polynomial : ℝ N → ℝ that is a continuation of B i in the sense that for all .
We now show that indeed satisfies the three requirements for interpolation functions that we set out at the beginning (see section on continuous homologues of Boolean functions).
Theorem. The function has the following properties:
- (i)
for all .
- (ii)
is the unique minimal degree polynomial interpolating B.
- (iii)
Let s denote the number of symmetry hyperplanes of B, i.e. the number of variables x i satisfying
B(x1, ..., xi-1, 0, xi+1, ..., x N ) = B(x1, ..., xi-1, 1, xi+1, ..., x N ), for all (x1, ..., , ..., x N ) ε {0, 1}N-1. Then deg ( ) = N - s.
- (iv)
is affine multilinear. It is multilinear iff it corresponds to an AND gate, i.e.B(x1, x2, ..., x N ) = .
- (v)
Let . Then it holds
In particular, if B is a Boolean function.
(ii) A minimal degree interpolation polynomial is of the form
since exponents greater than 1 can be replaced by 1 without changing the values of f for . We order the vertices of the unit cube such that the number of 1's in the coordinates is not decreasing and denote the reordered sequence by . Then we consider the sequence of equations f(V k ) = B(V k ), k = 1, 2, ..., 2 N . For all k ∈ {1, 2, ..., 2 N } there is exactly one coefficient whose monomial satisfies g(V k ) = 0, k = 1, 2, ..., k - 1 and g(V k ) = 1. This allows to uniquely determine the coefficients and that way also the polynomial f. Since is of the form (*), it is the unique minimal degree interpolation polynomial.
(iii) The degree of is clearly deg ( ) ≤ N. If B is symmetric with respect to the hyperplane x i = 0.5, w.l.o.g. i = 1, we have B(0, x2, ..., x N ) = B(1, x2, ..., x N ) for all (x2, ..., x N ) ∈ {0, 1}N-1and consequently
Hence, deg ( ) ≤ N - 1. Inductively this proves (iii).
(iv)Let i ∈ {1, 2, ..., N}. Then at fixed the derivative is constant so is affine linear in . Moreover, we have
For the last equivalence note that if is multilinear, then if any = 0.
(v) Assume and w.l.o.g. . Then it follows from (iv) that or . Inductively, we obtain x ∈ {0, 1} N such that , a contradiction to (i). Analogously, one proves that . □
This function does not have any symmetry hyperplanes, but its interpolation has degree 2.
Theoretical results about steady-states
The following theorem investigates the steady-state behavior of discrete and continuous models.
Theorem. Assume we are given a Boolean model and perfect continuous homologues of the Boolean update functions. Then for any state vector the following are equivalent:
(i) is a steady-state of the Boolean model.
(ii) is a steady-state of the model (1).
(iii) is a steady-state of the ODE model (2).
for model (2). Considering that the are perfect homologues of the B i , these conditions are clearly equivalent. □
Remark. Note that the above theorem generally applies to transformations using perfect continuous homologues of Boolean functions. In the special case of piecewise linear ODEs, i.e. from equation (6), Glass et al. [11] could show that the above theorem also holds when (iii) is replaced by ' is a stable steady-state of the ODE model (2)'.
We now extend the above theorem to HillCube models. The problem is that we can no longer assume a perfect agreement between and B i . The main idea is to use the implicit function theorem to prove the existence of a steady-state of the continuous model in a neighborhood of a Boolean steady-state.
Then there are open neighborhoods U1, U2 of 0 and 1, respectively, such that f( , m) is continuous on U1/2 × ℝ and it can be easily shown that f is even continuously differentiable on U1/2 × ℝ. Now consider the HillCube model. It depends only on the Hill exponents n and we define m = (m ij ) by . For concentrations let Φ(m, ) denote the right hand side of the HillCube ODE system. As explained above, we continuously extend the Hill functions and hence also Φ at m = 0. Then there exists an open neighborhood U ⊂ ℝ N of x∞ such that Φ is continuously differentiable on ℝ|m|× U (as the composition of continuously differentiable functions).
where δ ij is the Kronecker delta. Note that for m = 0 the Hill functions in the HillCubes are step functions, i.e. constant in a neighborhood of x∞. Hence the derivative of the HillCubes vanishes. This shows that DΦ is diagonal negative definite and hence, in particular, invertible. Therefore, the implicit function theorem guarantees that there are open neighborhoods U'of x∞ and V' ⊂ ℝ|m|of 0 such that for each m ∈ V', i.e. for large exponents n, there is a ∈ U' such that Φ(m, ) = 0, i.e. is a steady-state of the model. Since the mapping m ↦ is continuous we have → x∞ as m → 0 with respect to the euclidean norm, i.e. all m ij → 0 or equivalently n ij → ∞. Moreover, it follows that on subsets V ⊆ V' and U ⊆ U' the Jacobian of Φ is still negative definite and, consequently, the are stable steady-states. □
Manual parameter determination for the continuous T-cell model
For the activation of ZAP-70 k = 0.3 and τZAP-70= 10 are used. LAT phosp and cCbl are activated at thresholds kfast = 0.1 and kslow = 0.8, respectively, and both their life-times are set to τ = 1. Initial condition for all species is 0 and the inputs for ZAP-70 are fixed at 1. The result of the numeric simulation is shown in Figure 2D. From the beginning on ZAP-70 is activated but rather slowly due to the increased life-time τZAP-70. The activations of LAT-phosp and cCbl occur at around the time points when the concentration of ZAP-70 crosses the thresholds kfast and kslow, respectively. The high threshold kslow also leads to a lower total activation level of cCbl. One can clearly see the time lag between the activations of LAT phosp and cCbl, i.e. between the interactions in scenario 2 and the interactions not occurring until scenario 3.
Model transformation and simulation
For the transformation of the Boolean T-cell model we choose HillCube ODEs. HillCubes are better suited to describe signaling cascades than BooleCubes. Normalization is not necessary as we will choose thresholds k ≪ 1 and consequently the Hill functions will already satisfy f(1) ≈ 1. For the transformation and simulation we developed a MATLAB toolbox called Odefy [Krumsiek et al.: Odefy — From discrete to continuous models. In preparation (2009)], which is publicly available at http://cmb.helmholtz-muenchen.de/odefy and allows the experimentalist to easily transform Boolean models into ODE models. Since Odefy can be integrated into CellNetAnalyzer [24], we were able to export the Boolean T-cell model from there. Numeric integration of the ODE system was carried out using MATLAB ode15s, a variable-order multistep solver based on the numerical differentiation formulas.
Experimental data
Kemp et al. [12] created a data set describing the dynamics of the activation of the key signaling elements ERK, JNK, IKK and NFAT upon activation of the TCR. The data was generated by stimulation of a T-cell line (1B6 T cell hybridoma) with three peptides with different affinities for the T-cell receptor, Q144, Y144 and L144. In the case of L144 experiments were conducted for three different peptide concentrations 0.04 μ g/ml (low), 0.4 μ g/ml (medium) and 4 μ g/ml (high). In the case of Q144 and Y144 only the high concentration of 4 μ g/ml was used. Concentrations of ERK, JNK, IKK and NFAT were measured at 0, 10, 30, 60, 120, 240 and 2400 minutes. Here we neglected the last time point, as on this slow time scale many interactions play a role, that are not included in the model, such as gene expression. For the parameter fitting the data were linearly rescaled to the unit interval.
Parameter fitting
The T-cell model consists of 40 species, 55 pairwise interactions and three external inputs. Hence, we have 40 life-time parameters and 58 pairs of Hill parameters, amounting to a total of 156 parameters. Due to this large number of parameters compared to the number of experimental data points, the fitting problem is obviously ill-posed as for many different parameter sets the model reproduces the data equally well. For this reason, we performed a two-step fitting process. First, we determined a parameter set for which the model fits the experimental data reasonably well. Second, we added a regularization to account for the indeterminacies. Both steps are optimization problems. The two cost functions are given below. They take a parameter set consisting of all Hill parameters (n, k) and all life-times τ as input a yield a scalar loss value, that needs to be minimized. In both steps we used a simulated annealing algorithm [25] for minimization. As the threshold parameters k have to be precisely adjusted at small values, these were fitted on a log-scale, as is also done in [26]. Parameters downstream of the measured species were, of course, not changed but fixed at their manually determined value. We used the SBPD package of the Systems Biology Toolbox [27] to create a compiled MATLAB simulation function of our ODE model for faster performance.
Least squares fitting
where denotes the time courses predicted by the model and the interpolated data points. The simulated annealing algorithm was started from the manually determined parameter set and finally converged at a cost function value of 18.98.
where the last term is a penalty term ensuring a constant quality of the model's fit. The simulated annealing was started from the result of the first step. Idea of this regularization is to account for parameter indeterminacies by reducing their variation and to enhance the significance of 'outliers' like the affinity kL 144of the TCR for the ligand L144, cf. Figure 4E.
Additional data file 6 shows model simulations for 5 different results of step 1. While not perfectly agreeing, their overall dynamic behavior is the same. We can reasonably assume that neither the parameter indeterminacies nor the regularization significantly influence the dynamics of the experimentally observed species.
The authors would like to thank Melissa Kemp for providing the experimental data, Melody Morris for critical reading of the manuscript and useful discussions as well as the anonymous reviewers for their helpful comments during the preparation of this manuscript.
This research was partially supported by the Initiative and Networking Fund of the Helmholtz Association within the Helmholtz Alliance on Systems Biology (project CoReNe), by the German Federal Ministry of Education and Research (MaCS, Magdeburg Centre for Systems Biology) and by the Ministry of Education of Saxony-Anhalt (Research Center "Dynamic Systems"). JSR and DAL thank funding of Pfizer Inc., NIH grant GM68762, and DOD Institute for Collaborative Biotechnologies.
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This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

subset: commoncrawl


text:
Published: Tuesday, 10th July 2012 15:47 PM
Despite the summer month of July, the recent rainy weather has inspired a warm and cosy looking quilt of the month!
This striking yellow and white cotton wholecloth quilt was made by Frances Binns (nee Murray) from Corbridge, near Hexham, in 1902. It contains traditional North Country quilting patterns, including plaits, fan shapes and running feather motifs. Wholecloth quilting, where the emphasis of design focuses on the quilting patterns rather than patchwork piecing, is traditional to Wales, the North Country and the Scottish Borders. Each region has its own style and iconic quilting motifs that identify the quilt to its area of origin.
Unlike so many of the quilts in our collection, we have some information as to the maker of this beautiful piece. Frances Murray was born in 1851, and was rumoured to have come from a wealthy family background, but her mother had eloped with a coachman. Both of her parents died when she was in her teens. Together with her youngest sister Isabelle, they went to work as governess and cook to a family in Birkhamshaw, where Frances met her future husband, Henry Binns. They had four children, and her eldest son Frank went on to married fellow teacher Annie Hannam. Frances was very fond of Annie, and made her this quilt as a 21st Birthday present. It is thought Frances made a lot of quilts to supplement the family income and as gifts for different members of the family, although it is not certain what happened to these other quilts.
Interestingly the overall design of the quilt closely echoes another quilt in the Museum collection. The 'Triple X' quilt, also made in yellow and white cotton with a central 'X' design is thought to have been made by quilt teacher Nellie Ellison in the Castleside area of County Durham around 1910. In both cases, the quilting patterns have been drawn into the quilts using blue pencil, and traces of this can still be seen. Drawing quilting designs required a great deal of skill, and whilst both Frances and Nellie would have drawn their designs themselves, quilt tops could be sent away to have their designs drawn on by professional quilt marker.
White and yellow wholecloth quilt made by France Binns, 1902
'Triple X' Wholecloth Quilt, made by Nellie Ellison c.1910
The Quilters' Guild Collection is held by The Quilters' Guild of the British Isles.
Registered charity no 1067361 Company limited by guarantee registration no 34476331 Charity in Scotland no. SC043174.
© 2019 Quilt Museum and Gallery, York | Printed from: quiltmuseum.org.uk/blog/quilt-history-articles/01386.html


text:
A quick post to let you know there is another Mission Maxi sewing pattern giveaway. Natalie at Hungry Hippie is hosting the giveaway. Here's the link.
Also, the above pic is a sneak peek at the new pattern to come...
The new pattern looks awesome. Can wait to see more!
wow! the new pattern looks awesome. i can't wait to see more!
I have been looking for a pattern for the dress in your photo for a while. I can't wait to buy it!
Hi Jamie!
Hi Gloria!
I got a twitter account. It's gloriamft.
It won't actually have success, I consider this way.


text:
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What is "The Soul"
Exploring The Concept Of The Human Soul
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Subject: Exploring The Concept Of The Human Soul
Wed Nov 15, 2017 6:47 am
Exploring The Concept Of The Human Soul
The soul is your link with the divine, yet exactly what it is and how it works has been the subject of much theological and philosophical contention. The question "what is the soul?" has puzzled man for thousands of years, and many different religions and philosophers have all come up with their own view on this fundamental question. The main point of agreement is the the soul infuses the body and is the spark of life and consciousness that makes you the unique individual that you are. The mental processes called the "mind" are a part of the soul, but not its entirety.
A belief in the soul implies a belief in your own imortality, that although the physical flesh may perish, the spirit or essence, the soul, continues to exist beyond death. Where the soul goes, be it an afterlife, to reincarnate, or become one with divinity, is another question whose answer depends upon your spiritual beliefs.
There are certain philosophical terms regarding the soul, listed below:
Plurity of the soul:
The belief that the soul is comprised of parts.
Limited immortality:
The idea that the soul merges into oneness with the universe, losing indivuality to become part of a greater whole.
Another name for reincarnation, the transition of the soul from body to body through different lifetimes, in an effort to evolve to achieve perfection.
The journeying of the soul from one body to another, as in recarnation.
Universal Soul:
The concept that there is one perfect soul, from which all human souls begin and seek to reunite.
Twin soul:
The concept that two people share one soul.
Humanity has specualted on the nature of its divinity throughout the ages and across the world.
Here are some, but by no means all, of these ideas.
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Subject: Re: Exploring The Concept Of The Human Soul
Wed Nov 15, 2017 6:49 am
Ancient Greece
The Homeric poems, with which most ancient writers can safely be assumed to be intimately familiar, use the word 'soul' in two distinguishable, probably related, ways. The soul is, on the one hand, something that a human being risks in battle and loses in death. On the other hand, it is what at the time of death departs from the person's limbs and travels to the underworld, where it has a more or less pitiful afterlife as a shade or image of the deceased person. It has been suggested that what is referred to as soul in either case is in fact thought of as one and the same thing, something that a person can risk and lose and that, after death, endures as a shade in the underworld. The suggestion is plausible, but cannot be verified. In any case, once a person's soul has departed for good, the person is dead. The presence of soul therefore distinguishes a living human body from a corpse. However, this is plainly not to say that the soul is thought of as what accounts for, or is responsible for, the activities, responses, operations and the like that constitute a person's life. Homer never says that anyone does anything in virtue of, or with, their soul, nor does he attribute any activity to the soul of a living person. Thus, though the presence or absence of soul marks out a person's life, it is not otherwise associated with that life. Moreover, it is a striking feature of Homeric usage that, in Furley's words, to mention soul is to suggest death: someone's soul comes to mind only when their life is thought, by themselves or others, to be at risk. Thus Achilles says that he is continuously risking his soul (
9.322), and Agenor reflects on the fact that even Achilles has just one soul (
It should also be pointed out that in the Homeric poems, only human beings are said to have and to lose souls. Correspondingly, Homer never envisages shades or images of non-human creatures in the underworld. These two facts taken together suggest that in whatever precise way the soul is conceived of as associated with life, it is in any case thought to be connected not with life in general, or life in all its forms, but rather, more specifically, with the life of a human being.
Several significant developments occurred in the ways Greeks thought and spoke about the soul in the sixth and fifth centuries. The questions about the soul that are formulated and discussed in the writings of Plato and Aristotle to some extent arise from, and need to be interpreted against the background of, these sixth and fifth century developments. One factor that is of central importance is the gradual loss of the Homeric connection between mentioning a person's soul and the thought that their life is vulnerable or at risk.
In ordinary fifth century Greek, having soul is simply being alive; hence the emergence at about this time, of the adjective 'ensouled' [
] as the standard word meaning "alive", which was applied not just to human beings, but to other living things as well. There is some reason to think that the word 'soul' was used in this straightforwardly positive way already in the sixth century. Thales of Miletus, who is credited with successfully predicting a solar eclipse occurring in 585, reportedly attributed soul to magnets, on the grounds that magnets are capable of moving iron (Aristotle,
De Anima
1.2, 405a19-21). Thales' thought was presumably that since it is distinctive of living things to be able to initiate movement, magnets must in fact be alive or, in other words, ensouled. Thus, while Homer spoke of soul only in the case of human beings, in sixth and fifth century usage soul is attributed to every kind of living thing. What is in place, then, at this time is the notion that soul is what distinguishes that which is alive from that which is not.
However, it is not just that soul is said to be present in every living thing. It is also the case that an increasingly broad range of ways of acting and being acted on is attributed to the soul. Thus it has come to be natural, by the end of the fifth century, to refer pleasure taken in food and drink, as well as sexual desire, to the soul. People are said, for example, to satisfy their souls with rich food, and the souls of gods and men are claimed to be subject to sexual desire. In contexts of intense emotion or crisis, feelings like love and hate, joy and grief, anger and shame are associated with the soul.
"Nothing bites the soul of a man more than dishonor"
, says Ajax in a fragment from a tragedy of unknown authorship, just before he commits suicide. Oedipus says that his soul laments the misery of his city and its inhabitants (
Oedipus Tyrannus
64). Moreover, the soul is also importantly connected with boldness and courage, especially in battle. Courageous people are said, for instance in Herodotus and Thucydides, to have enduring or strong souls. In the Hippocratic text
Airs, Waters, Places
, the soul is thought of as the place of courage or, as the case may be, its opposite: in the case of lowland inhabitants, courage and endurance are not in their souls by nature, but must be instilled by law, similarly in benign climates, men are fleshy, ill-jointed, moist, without endurance and weak in soul.
The connection between the soul and characteristics like boldness and courage in battle is plainly an aspect of the noteworthy fifth century development whereby the soul comes to be thought of as the source or bearer of moral qualities such as, for instance, temperance and justice. In Pericles' funeral oration that Thucydides includes in his account of the Peloponnesian War, he says that those who know most clearly the sweet and the terrible, and yet do not as a result turn away from danger, are rightly judged
"strongest with regard to soul"
. This text, and others like it indicate a semantic extension whereby 'soul' comes to denote a person's moral character, often, but not always, with special regard to qualities such as endurance and courage. While the connection with courage is obvious in a number of texts, there are other texts in which the soul is the bearer of other admirable qualities, such as a Euripidean fragment that speaks of the desire characteristic of a soul that is just, temperate and good. Hippolytus, in Euripides' play named after him, describes himself as having a "
virgin soul
" (
1006), obviously to evoke his abstinence from sex. In Pindar's second Olympian, salvation is promised to those who "
keep their souls from unjust acts
" (2.68-70). The last two texts mentioned may well be influenced by Orphic and Pythagorean beliefs about the nature and immortality of the soul, to which we will turn in due course. But it would be a mistake to think that the moralization of the soul (i.e. its association with moral characteristics) wholly depended on Orphic and Pythagorean speculation. It would, at the very least, be to disregard the soul's connection with courage in poetry, the historians and in Hippocratic writings.
To educated fifth century speakers of Greek, it would have been natural to think of qualities of soul as accounting for, and being manifested in, a person's morally significant behavior. Pericles acts courageously, and Hippolytus temperately, or chastely, because of the qualities of their souls from which such actions have a strong tendency to flow, and their actions express and make evident the courage, temperance and the like that characterize their souls. Once we are in a position properly to appreciate the connection between soul and moral character that must already have been felt to be natural at this stage, it should come as no surprise that the soul is also taken to be something that engages in activities like thinking and planning. If the soul is, in some sense, responsible for courageous acts, for instance, it is only to be expected that the soul also grasps what, in the circumstances, courage calls for, and how, at some suitable level of detail, the courageous act must be performed.
Thus in a speech of Antiphon, the jury is urged to "
take away from the accused the soul that planned the crime
", in striking juxtaposition of the ideas of life-soul (as in Homer) and of soul as responsible for practical thought. Somewhat similarly, in a Sophoclean fragment someone says that "
a kindly soul with just thoughts is a better inventor than any sophist
" (cf. also Euripides,
1180). Moreover, it is easy to see that there are connections between familiar uses of 'soul' in emotional contexts and attributions to the soul of cognitive and intellectual activities and achievements. There is, after all, no clear-cut and manifest difference between, say, being in the emotional state of fear and having a terrifying thought or perception. When Oedipus' soul laments, or Ajax's soul is bitten by dishonor, emotion obviously goes hand in hand with cognition, and if it is natural to refer the one to the soul, there should be nothing puzzling about attributions to it of the other. Thus in non-philosophical Greek of the fifth century the soul is treated as the bearer of moral qualities, and also as responsible for practical thought and cognition.
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Subject: Re: Exploring The Concept Of The Human Soul
Wed Nov 15, 2017 6:51 am
Plato's Theories of Soul
The various developments that occurred in the sixth and fifth centuries in how Greeks thought and spoke of the soul resulted in a very complex notion that strikes one as remarkably close to conceptions of the soul that we find in fourth century philosophical theories, notably Plato's. There is thus some reason to think that the philosophical theories in question are best interpreted as working with, and on, the relatively non-theoretical notion of the soul that by the end of the fifth century has come to be embedded in ordinary language. In what follows our main concern will be to characterize some of the theories in question. But we should also attend, wherever this seems appropriate and helpful, to ways in which familiarity with the ordinary notion of the soul might enable us better to understand why a theory or an argument proceeds the way it does. In addition, we should note ways in which philosophical theories might seem to clarify and further articulate the ordinary notion. We begin with Plato, and with a question that is intimately tied up with the ordinary notion of the soul as it developed from the Homeric poems onwards, namely whether a person's soul does indeed survive the person's death.
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Subject: Re: Exploring The Concept Of The Human Soul
Wed Nov 15, 2017 6:56 am
Plato and the Soulmate Theory.
In the Symposium dated c. 385–370 BC, Plato talked about the theory that originally humans had 2 hearts, 2 heads and 4 legs. Some interpretations say that Zeus feared the power of humans and some say he was aggravated with human arrogance. So he split the soul in two. This created a longing for completeness that no other soul could fill. The two souls would go through lifetimes searching for their other half. In theory the point of twin flames is that eventually the two polarities of the same soul, holding the same frequency, will reunite breaking through the veil and remembering their true selves.
It's written throughout world mythology with slight variations.
Christianity teaches the immortality of the soul, which is created by God and placed on Earth to experience life and exercise free will. After death the soul will be judged on its deeds and go to heaven or hell accordingly. But what does it teach of soulmates?
Old Testament
In Judaeo-Christian soul mate theory, God fashions an adrogynous creature containing both sexes in his own image, which comprises the essence of spirit. God then decides to give this spirit a 'living soul' and creates Adam, and then his female 'half' Eve out of his rib.
New Testament
As in Plato's soul mate theory, the New Testament says that humans were was once whole, but were then divided to create its mate. Jesus reminds the Pharisees that God had originally made them:
Quote :
"male and female... for this cause shall a man leave father and mother and shall cleave to his wife, and they twain shall be one flesh... no more twain, but one flesh." – Matthew 19:4-6
Hindus believe that all souls are part of a Universal Soul, Atman, which is also the sum of all the Gods. The soul is the unity that links all living beings, as parts of the greater whole. The soul transmigrates through different incarnations until it realises its unity with Atman and no longer needs to reincarnate. In Hinduism soul mate theory, the universal soul becomes conscious of itself, desire companionship, and therefore brings forth from its own Being the male and the female.
Quote :
"He then made his Self fall in two, and thence arose husband and wife..." – Robert O. Ballou, The Portable World Bible
The legend of the Egyptian Gods
Osiris and Isis
dates back 5,000 years. Essentially, these two began their connection in the womb, where they are born as twins. They're also very much in love. Later in life, Osiris is kidnapped and killed by his jealous brother, Set. In grief, Isis merges with Oriris' spirit, and they conceive a god-like child, Horus. Angered, Set has his brother's body cut up into fourteen pieces. In response, Isis shows her eternal love by gathering the pieces of her husband's body, until he eventually comes back to life.
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Subject: Re: Exploring The Concept Of The Human Soul
Wed Nov 15, 2017 7:00 am
Aristotle's Theory of Soul
Aristotle's theory, as it is presented primarily in the
De Anima
, comes very close to providing a comprehensive, fully developed account of the soul in all its aspects and functions, an account that articulates the ways in which all of the vital functions of all animate organisms are related to the soul. In doing so, the theory comes very close to offering a comprehensive answer to a question that arises from the ordinary Greek notion of soul, namely how precisely it is that the soul, which is agreed to be in some way or other responsible for a variety of things living creatures (especially humans) do and experience, also is the distinguishing mark of the animate. According to Aristotle's theory, a soul is a particular kind of nature, a principle that accounts for change and rest in the particular case of living bodies, i.e. plants, nonhuman animals and human beings. The relation between soul and body, on Aristotle's view, is also an instance of the more general relation between form and matter: thus an ensouled, living body is a particular kind of in-formed matter. Slightly simplifying things by limiting ourselves to the sublunary world we can describe the theory as furnishing a unified explanatory framework within which all vital functions alike, from metabolism to reasoning, are treated as functions performed by natural organisms of suitable structure and complexity. The soul of an animate organism, in this framework, is nothing other than its system of active abilities to perform the vital functions that organisms of its kind naturally perform, so that when an organism engages in the relevant activities (e.g., nutrition, movement or thought) it does so in virtue of the system of abilities that is its soul.
Given that the soul is, according to Aristotle's theory, a system of abilities possessed and manifested by animate bodies of suitable structure, it is clear that the soul is, according to Aristotle, not itself a body or a corporeal thing. Thus Aristotle agrees with the
's claim that souls are very different from bodies. Moreover, Aristotle seems to think that all the abilities that are constitutive of the souls of plants, beasts and humans are such that their exercise involves and requires bodily parts and organs. This is obviously so with, for instance, the abilities for movement in respect of place (e.g., by walking or flying), and for sense-perception, which requires sense-organs. Aristotle does not, however, think that there is an organ of thought, and so he also does not think that the exercise of the ability to think involves the use of a bodily part or organ that exists specifically for this use. Nevertheless, he does seem to take the view that the activity of the human intellect always involves some activity of the perceptual apparatus, and hence requires the presence, and proper arrangement, of suitable bodily parts and organs; for he seems to think that sensory impressions [
] are somehow involved in every occurrent act of thought, at least as far as human beings are concerned. If so, Aristotle in fact seems to be committed to the view that, contrary to the Platonic position, even human souls are not capable of existence and (perhaps as importantly) activity apart from the body
It is noteworthy that Aristotle's theory does not mark off those vital functions that are mental by relating them to the soul in some special way that differs from and goes beyond the way in which vital functions in general are so related. It is certainly not part of Aristotle's theory that the soul is specially and directly responsible for mental functions by performing them on its own, whereas it is less directly responsible for the performance by the living organism of other vital functions such as growth. As this aspect of his theory suggests, Aristotle is confident that once one has a proper understanding of how to explain natural phenomena in general, there is no reason to suppose that mental functions like perception, desire and at least some forms of thinking cannot be explained simply by appealing to the principles in terms of which natural phenomena in general are properly understood and explained.
It might be thought that since Aristotle's theory treats mental functions and other vital functions exactly alike, it obscures a crucial distinction. This worry, however, turns out to be unjustified. The theory treats mental and other vital functions alike only in that it views both kinds of functions as performed by natural organisms of the right kind of structure and complexity. Viewing mental and other vital functions in this way is perfectly compatible with introducing a distinction between mental and other functions if concerns of some kind or other call for such a distinction. Aristotle is perfectly capable, for instance, of setting aside non-mental vital functions as irrelevant for the purposes of practical philosophy.
Ancient philosophy did not, of course, end with classical Stoicism, or indeed with the Hellenistic period, and neither did ancient theorizing about the soul. The revival of interest in the works of both Plato and Aristotle beginning in the second half of the second century B.C. prominently included renewed interest in Platonic and Aristotelian conceptions of the soul, sparking novel theoretical developments, such as, for instance, Plotinus' argument that the soul could not be spatially extended, since no spatially extended item could account for the unity of the subject of sense-perception. Christian writers such as Clement of Alexandria and Gregory of Nyssa were heavily indebted to philosophical theories of soul, especially Platonic ones, but also introduced new concerns and interests of their own. Nevertheless, these and other post-classical developments in every case need to be interpreted within the framework and context furnished by the classical theories that some of us have been considering in some detail.
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Subject: Re: Exploring The Concept Of The Human Soul
Exploring The Concept Of The Human Soul
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Spring and warmer weather means festivals and events. Some, hosted by pet shelters, focus on our four-legged companions. One such event (aptly-named Woofstock) is taking place near Atlanta next weekend. These types of events are held across the country in a variety of forms, but they have certain elements in common: pet parades and pet costume contests, food, and music. Some add agility games while others make it a weekend-long event with all types of entertainment. Most benefit a local shelter or pet rescue group. And, well, they are a lot of fun.
I'm a big fan of anything that helps animals, especially those in need of a home. About a month ago, a friend of mine adopted a dog that had been abandoned. He was discovered and taken to a local rescue, where my friend found his picture on their Web site. When he was finally healthy enough for her to bring home, I stopped by to see her new addition. He looked so fragile, and all of his ribs were visible through his shaved coat, despite the fact that he had gained enough weight to be sent home with her. But in spite of all his troubles, he's the sweetest little dog and I was heartbroken by his sad story. A few days ago, I dropped in to see him again and I couldn't believe how far he has come in just a few weeks. He has gained weight, he's happy and full of personality, and he has bonded strongly with my friend's daughter, quickly becoming a valued family member. Unfortunately, a lot of dogs and cats in shelters aren't so lucky.
Perhaps you've been considering a dog or cat – many apartment complexes are very welcoming to animals, even offering private dog parks for residents. If you live in a pet-friendly apartment community and are considering adding a dog or cat to your home, consider adoption. If you don't know where to start, try the Humane Society or the ASPCA. From these Web sites, you'll be able to search for pets available for adoption, get advice on pet-proofing your home, find out what you need to do before you adopt, and get tips on the pet adoption process.
First, consider your living space. If you don't know, find out your apartment complex's rules on pets. Some places are cat only; others allow dogs up to a certain size. Others may have breed restrictions. You'll also want to ask about fees – is there a pet fee? If so, is it a one-time pet deposit or is there a monthly charge? Based on the answers, you will be able to decide if a pet will fit into your current lifestyle. If so, then will a dog or a cat work better for you? If it is a dog, you can then decide on the size of the dog.
If you've decided on a dog, research the breeds depending on the size of dog you are looking for. Small dogs, for example, can range from the tiny and active, such as the chihuahua, to the rotund and lazy (sorry, Peanut) pug. Some (such as pugs) are great with children; but some small breeds are better off with just adults. Select based on your family and lifestyle. It is a myth that you can't rescue a pure-breed or a puppy, but don't overlook the mixed breeds and older dogs. My friend's dog is nearly two years old and appears to be mostly schnauzer, but he's apparently part poodle. This combination gives him an adorable mixed-up look, with his schnauzer face and curly poodle coat.
Once your home is ready and you've stocked up on supplies, it's time to bring home your new best friend. Rescuing a pet adds another level of satisfaction and happiness to pet-ownership. Enjoy being a hero to your new little roommate – you're both on your way to a whole new adventure.


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Quick Links
Canada's official languages are English and French.
Upon arrival to Canada you may need to sign up for English classes. In order to do this you must go to the Language Assessment and Resource Centre to have your language levels assessed. This is done through a written and oral test. After you do this, you will know what classes you are able to take.
The Language Assessment and Resource Centre will be able to direct you to the organizations that offer the classes that you are able to take.
These centres may have Care for Newcomer Children services available for you during class times. Contact the centre that you will be taking classes at to learn more about their Care for Newcomer Children services. Many of these locations offer classes at different times during the day so you are able to find one that will work best for you.
There are a variety of classes that have different focuses and different levels of ability. These are some of the classes that you may see in Windsor-Essex.
The English as a Second Language (ESL) program is offered by the Greater Essex County District School Board. In order to use this program you need to:
If you are a visitor to Canada, on a Student Visa or an Individual on Work Authorization, you are able to attend ESL classes but you may need to pay additional fees.
The LINC program is a language training program that is funded by Immigration, Refugees and Citizenship Canada. The program is free, but you need to:
Canadian citizens, refugee claimants and temporary residents are not allowed to take LINC.
Enhanced Language Training is job-specific and helps you learn the language that is used in the field you hope to be working in.
The Conseil scolaire catholique Providence offers French classes in most of its elementary and secondary schools, in the Windsor, Essex, Chatham-Kent and Sarnia areas. Visit their website to learn more about these classes.
You may be interested in learning French as a Second Language in a school setting. These courses have an additional cost to them.
You may learn more about these courses through Collège Boréal or St.Clair College.


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For the first time in its 150-year history, Wheaton College has formally instated a community group for same-sex attracted students. "Refuge" was recognized as a community group under the Student Care Committee for Student Government by President Philip Ryken Tuesday, and is composed of approximately 15 Wheaton students who identify as same-sex attracted.
According to a campus-wide email sent by President Ryken this week, all of Wheaton's community groups under the Student Care Committee are meant to "bring students together for confidential support by others who share a similar life experience. They are not therapy groups, but they are venues to care for students, including students who feel isolated and have not found a place to connect in the Wheaton Community."
The group is private and confidential, with an application process required for admittance. For the past 18 months, the group has been meeting with the Dean of Student Care and Services Melanie Humphreys, and has grown from two students to 15. According to Humphreys, "Refuge is one of the most at-risk student groups on campus…and what I mean by 'at-risk' is at risk for self-harm or suicide. Each of the students I have come to know have experienced significant loneliness and isolation on our campus."
"Refuge saved (my roommate's) life, and it probably saved mine, too," a current Wheaton student participating in Refuge said.
For more information on Refuge, and other groups like it at Christian colleges around the country, read "Hope for the Gay Undergrad," a feature article I wrote for Christianity Today magazine last month. The Wheaton Record, Wheaton's campus newspaper, also covered the history of the school's same-sex attracted hospitality in a cover story published online today at this link: "Refuge Becomes an Official Group for Students Questioning their Gender Identity or Sexual Orientation."
According to President Ryken, "Refuge" was chosen as the name for the group because: "There are many references to divine refuge in Scripture, such as the first verse of Psalm 46: 'God is our refuge and strength, an ever-present help in trouble.' The goal of Refuge is to provide support for students in their journey to understand their identity in Christ and its implications for their sexual orientation and gender identity."
Today at 3:30 p.m., Wheaton will welcome Andrew Marin, founder of The Marin Foundation, a Chicago-based non-profit focused on building bridges between the LGBT community and the Church, to campus for a presentation at Barrows Auditorium.
Filed under: Christian Life 101, Profile, Truth Bomb
Tags: homosexuality, refuge, same sex attraction, the marin foundation, wheaton college


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Amanda Lee, PA-C
Amanda Warren Lee is a graduate of Meredith College where she earned a Bachelor of Science degree in Biology with Pre-Health Concentration and a Bachelor of Arts in Mathematics. She later graduated from Methodist University, earning a Master of Medical Science degree in Physician Assistant studies. She has worked in the area of emergency medicine as well as urgent care. She is a NCCPA board-certified Physician Assistant and is a member of the NC Medical Board. She provides medical care and well care for pediatrics, adolescents, and adults.


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Concentrating on their minute project.
I had a great time and this event encouraged me a lot. Thank you so much for your help Junk Evolution, Char and all the students. I hope to see you again soon!
Congrats to you, Hiromi, for your successful first craft class in the U.S.!! Yippee!! And oh, that white & silver ring looks just so lovely!ReplyDelete
Thank you so much Amy! I am so excited to teach again. Having events as an artist in this area makes me feel to be connected to this community.
The girl who made a W&S ring just got engaged recently. I think the ring suits her perfect!
That looks a fun class! I wish I could join in your class:( I hope you have more classes soon.ReplyDelete
Thank you PortulaKa!
I wish I could have my good online friends in my class! That would be so much fun to talk about all kinds of crafting! I am going to have Furoshiki demo tomorrow.


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This article was published in the January issue of Blue Skies Magazine.
The Six Traits of Highly Successful DZO's
After more than four years of working with DZO's from every size DZ, I have noted six distinct characteristics associated with a DZ's health both financially and culturally.
The six traits of the most successful DZO's exhibit the following:
- Self-Awareness
- Clear Vision
- Builds Strong Teams
- Focuses on the Big and Small Details
- Data Driven
- Thick Skinned
Too many DZO's have their hands in too many things. As a startup DZ, this can be good as it keeps expenses down, but as the business scales, a team must be created; otherwise, burnout is assured.
If a DZO is burnt out and mentally exhausted, then it's a battle for that business to reach its potential. The success of a DZ is predicated by a DZO's self-awareness to recognize mental exhaustion and to actively step away from their business to maintain needed objectivity to handle the strains of running a drop zone.
No one is immune from high levels of stress for extended periods of time. Stress kills the passion and attention to detail begins to fade followed by a DZ's culture.
The DZO is the life force, and that force needs to be positive and healthy for the operation to grow. Those with self-awareness understand this and take the necessary steps to maintain perspective.
A Clear Vision
The best DZO's have a vision for the organization clearly expressed as well as a list of core values shared with all.
This vision is shared from the DZO and is clearly defined. In other words, what is the moral compass for the operation? What is the modus operandi? What are the DZO's expectations? Without a roadmap or guidepost, the journey will be rough.
When everyone shares a common goal and shared values, the organization is a more fulfilling place for all.
Builds a Strong Team
Too many DZO's have their hands in too many things. As a startup DZ, this can be good as it keeps expenses down, but as the business scales, a team must be created; otherwise, burnout is assured.
The best DZO's recognize that they can't do it all and must begin to build a team based on both skill and personality.
Focuses on the Big and Small Details
The best marketing begins when people arrive at a business, not just what you did to get them there.
DZO's that understand the value of creating an experience beyond the skydive generate a powerful marketing undercurrent. This is an investment into a business' long-term survival and is something that DZ staff want to be part of.
The best DZO's recognize that it's minute details that create a great experience far beyond the skydive itself.
Data Driven
Numbers don't lie, and the best DZO's know them inside and out from the actual cost of a tandem to statistics on Google. Having a good grasp of the numbers allows a DZO to make sound decisions for the business.
Thick Skinned
Everyone has an opinion, and many aren't afraid of expressing how a DZO should run a business. With thoughts, ideas and criticisms coming in from all sides, plus the financial strains of having to fix very expensive things, a DZO has to be very thick skinned to make it in this business.
Notice a Trend?
All of these traits combat a theme mentioned several times in this article- burnout. If I were to write a DZO handbook, it would highlight this issue much more. This topic is seldom talked about but is an issue that every skydiving operator must address if longevity is a desire.
Have questions? E-mail me! firstname.lastname@example.com
Upcoming Events
- 2018 DZ Marketing Conference – February 6-7, 2018
4 thoughts on "Six Traits of Highly Successful DZO's"
A good dzso need to be intuitive and have the ability to very quickly THINK calmly while running. So often situations arise at the dz either between staff, or staff and client that can escalate into an uncomfortable situation. It can result in a negative vibe and atmosphere developing at the dz or reflect poorly on the dz as a while. The dzso need to be quickly observant of this and be able to"nip"things in the bud It may also mean that the dzso is able to defer negative outcomes from these situations to a more suitable time of the day.


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I've got a pig obsession. Even if I attempted to explain the story to you, it would never make sense and you would seriously think I am some sort of crazed person. All you need to know is the love of pigs comes from an inside joke between my sister and I and now she calls me pig.
I literally squealed for joy when I saw this little Gucci accessory pop up on Net-A-Porter. Some kitschy accessories are released from time to time and many of them are far too absurd for me to ever want to splurge on them. But today I've got a major itch for the Gucci Little Pig Leather and Canvas Coin Purse and I plan to buy one for me and one for my sister. (more…)